recursal/Fanatic-Fandom huggingface dataset:
Resistors in Series and Parallel

## **Problem**

Consider three resistors with different resistivity

R
1

{\displaystyle R\_1}

,

R
2

{\displaystyle R\_2}

 and

R
3

{\displaystyle R\_3 }

. The formula for calculating the voltage across a resistor is

# V
I
R

{\displaystyle V = IR}

.

**Part 1:** Determine the equivalent resistance when the three resistors are connected in series.

**Part 2:** Determine the equivalent resistance when the three resistors are connected in parallel.

## **Solution**

**Part 1: Resistors connected in series (same current, different voltage)**

Let

V

{\displaystyle V}

 represent the voltage of the battery that follows Ohm's rule

# V
I

R

e
q

{\displaystyle V = I{R}\_{eq} }

. Here

R

e
q

{\displaystyle {R}\_{eq} }

 represents the equivalent resistance of the resistors. When two resistors are connected in series, the current going through each resistor is the same, and the voltage drop across each resistor is proportional to their respective resistances. By Kirchoff's loop rule,

V
−

I

1

R

1

−

I

2

R

2

−

I

3

R

3

=
0

{\displaystyle V - {I}\_{1}{R}\_{1} - {I}\_{2}{R}\_{2} - {I}\_{3}{R}\_{3} = 0 }

# V

I

1

R

1

I

2

R

2

I

3

R

3

{\displaystyle V = {I}\_{1}{R}\_{1} + {I}\_{2}{R}\_{2} + {I}\_{3}{R}\_{3} }

I

R

e
q

=

I

1

R

1

I

2

R

2

I

3

R

3

.

{\displaystyle I{R}\_{eq} = {I}\_{1}{R}\_{1} + {I}\_{2}{R}\_{2} + {I}\_{3}{R}\_{3}. }

Since

I
1

=

I
2

=

I
3

=
I

{\displaystyle I\_1 = I\_2 = I\_3 = I }

,

I

R

e
q

=
I

(

R

1

R

2

R

3

)

{\displaystyle I{R}\_{eq} = I\left({R}\_{1} + {R}\_{2} + {R}\_{3}\right) }

R

e
q

=

R

1

R

2

R

3

.

{\displaystyle {R}\_{eq} = {R}\_{1} + {R}\_{2} + {R}\_{3}. }

In general, given

n

{\displaystyle n}

 resistors added in series

R

e
q

=

R

1

R

2

R

3

⋯

R

n

.

{\displaystyle {R}\_{eq} = {R}\_{1} + {R}\_{2} + {R}\_{3} + \cdots + {R}\_{n}. }

**Part 2: Resistors connected in parallel (same voltage, different current)**

When two resistors are connected in parallel, the voltage going through each resistor is the same, and the current the enters into the parallel branch is distributed across each resistor inversely-proportional to each resistor. By Kirchoff's junction rule,

# I

I
1

I
2

I
3

{\displaystyle I = I\_1 + I\_2 + I\_3 }

V

R

e
q

=

V

1

R

1

V

2

R

2

V

3

R

3

.

{\displaystyle \frac{V}{{R}\_{eq}} = \frac{{V}\_{1}}{{R}\_{1}} + \frac{{V}\_{2}}{{R}\_{2}} + \frac{{V}\_{3}}{{R}\_{3}}. }

Since

V
1

=

V
2

=

V
3

=
V

{\displaystyle V\_1 = V\_2 = V\_3 = V }

,

V

R

e
q

=

V

R

1

V

R

2

V

R

3

{\displaystyle \frac{V}{{R}\_{eq}} = \frac{V}{{R}\_{1}} + \frac{V}{{R}\_{2}} + \frac{V}{{R}\_{3}}}

1

R

e
q

=

1

R

1

1

R

2

1

R

3

.

{\displaystyle \frac{1}{{R}\_{eq}} = \frac{1}{{R}\_{1}} + \frac{1}{{R}\_{2}} + \frac{1}{{R}\_{3}}.}

In general, given

n

{\displaystyle n}

 resistors added in parallel

1

R

e
q

=

1

R

1

1

R

2

1

R

3

⋯

1

R

n

.

{\displaystyle \frac{1}{{R}\_{eq}} = \frac{1}{{R}\_{1}} + \frac{1}{{R}\_{2}} + \frac{1}{{R}\_{3}}+ \cdots + \frac{1}{{R}\_{n}}. }
Melissa

Full name: Melissa Kathryn Connell

Born: December 28, 1999 (Age 21)

## Additional info

* Occupation: Not known yet
* Height: 5'9
* Zodiac sign: Capricorn
* Favorite colors: Neon pink, orange, yellow, green, blue, purple and teal
* Handedness: Right
* Belly button: Innie
* Ticklish level: 9
* Shoe size: 9
* Toe color: Bright blue

## Trivia

* She is based off the announcer/host in Keep It Spotless, Melissa Vanderschyff.
* She graduated high school in 2018.
* She can do a good hillbilly impression (one example is from SpongeBob's Chocolate With Nuts where the hillbilly says, "I'll take 20")
* Her favorite pizza toppings are pepperoni, sausage and jalapenos.
* She likes to bake.
* She likes seafood. Crab cakes are one of her favorites.
Nasty Food Jenga

You need:

* A Jenga tower
* Red, orange, yellow, green, blue, and white paint
* A black permanent marker
* 12 buckets, 2 of each being red, orange, yellow, green, blue, and white
* 2 white lids
* A food for each bucket

## Setup

1. Separate the Jenga blocks into 5 groups of 10, and one group of 4.
2. Paint the first group of 10 blocks red, and repeat this step for the other groups of blocks using orange, yellow, green, and blue paint.
3. Paint the remaining 4 blocks white.
4. Let the blocks dry, then draw question marks on the sides of the white blocks because they represent the white mystery bucket.
5. Draw question marks on the white lids for the white buckets.
6. Arrange the painted blocks back in the form of a tower, placing each color randomly.
7. Place a different food in each bucket according to these rules:
8. Place a food the player really likes in the blue bucket.
9. Place a food that is edible but the player doesn't really like in the green bucket.
10. Modify another food that the player likes by putting something weird and/or disgusting in it and place it in the yellow bucket. (such as cat food Oreos, or marmite Twinkies)
11. Hand-make a weird, disgusting creation and place it in the orange bucket.
12. Place a food that's really nasty or bad in the red bucket.
13. Modify another batch of the food in the blue bucket by adding a non-toxic inedible item (such as glue or soap) and place it in the mystery bucket, and close it up with the white lid.
14. Have a trash can ready for each player in case they have to puke due to a disgusting/unknown food.

## How to play

1. Players will take turns pulling blocks out of the tower, and the player has to eat the food from the bucket the same color as the block pulled.
2. The player who topples the tower has to drink a smoothie of all their foods as a punishment for losing.
Clara

Full name: Clara Perrie

Born: October 22, 1991 (age 28)

A pretty lady in her 20s. Currently has a job at Jill's Bakery as a waitress/assistant, and likes talking to people in other tables. Likes bananas, nutty butter cups, and butterscotch topping. She has a ton of flip flops in a variety of colors, which are the only shoes she wears now (threw out her black sneakers with blue laces in December 2015). Her occupation is DDR player

## Additional info

* Occupations: DDR player, worker at Jill's bakery
* Height: 5'9
* Zodiac sign: Libra
* Handedness: Right
* Belly button: Innie
* Favorite foods: Bananas, nutty butter cups, butterscotch topping
* Ticklish level (1-10): 8.5
* Shoe size: 8.5
* Toe color: Pink
* Food day she was born on: National Nut Day
* Phone number: (592) 070-4065
Water Molecules in a Snowball

**Problem created by Tao Steven Zheng (郑涛)**

## **Problem**

How many water molecules

H

2

O

{\displaystyle \ce{H2O}}

 are in a fist-size snowball?. Give your answer as an *order of magnitude*.

This problem appeared in the 2013 UBC Physics Olympics Fermi problems event.

## **Solution**

**Rough Estimate**

Radius of a fist-size snowball:

5

c
m

{\displaystyle 5 \; cm }

Density of water:

1

g

/

c

m
3

{\displaystyle 1 \; g/cm^3 }

Molar mass of water (

H

2

O

{\displaystyle \ce{H2O}}

):

2
×
1

g

/

m
o
l
16

g

/

m
o
# l
18

g

/

m
o
l

{\displaystyle 2 \times 1 \; g/mol + 16 \;g/mol = 18 \; g/mol }

To calculate the number of water molecules in a fist-size snowball, use the density of water multiplied by the volume of the snowball, divided by the molar mass of 1 water molecule, multiplied by Avogadro's number (

6.02
×

10

23

m
o
l

−
1

{\displaystyle 6.02 \times {10}^{23} \; {mol}^{-1} }

). For the rough calculation, we will approximate Avogadro's number to be

6
×

10

23

m
o
l

−
1

{\displaystyle 6 \times {10}^{23} \; {mol}^{-1} }

.

1

g

c

m
3

×

12

(
5

c
m
)

3

3

×

m
o
l

18

g

×

6
×

10

23

m
o
l

=
1.67
×

10

25

{\displaystyle \frac{1 \; g}{cm^3} \times \frac{12{(5 \; cm)}^{3}}{3} \times \frac{mol}{18 \; g} \times \frac{6 \times {10}^{23}}{mol} = 1.67 \times {10}^{25} }

**More Accurate Estimate**

Density of ice:

0.917

g

/

c

m
3

{\displaystyle 0.917 \; g/cm^3 }

For a more accurate estimate, use the density of ice instead of water, and use the the actual Avogadro's number.

0.917

g

c

m
3

×

4
π

(
5

c
m
)

3

3

×

m
o
l

18

g

×

6.02
×

10

23

m
o
l

=
1.61
×

10

25

{\displaystyle \frac{0.917 \; g}{cm^3} \times \frac{4\pi{(5 \; cm)}^{3}}{3} \times \frac{mol}{18 \; g} \times \frac{6.02 \times {10}^{23}}{mol} = 1.61 \times {10}^{25} }

The order of magnitude is 25.
324 Entertainment Shorts

An animated short series produced by 324 Entertainment. Episodes can be divided into 2-5 segments.

Episodes usually air on Saturday nights, with a few exceptions here and there.

## Specials released before 324 Entertainment Shorts (usually featuring real Flipline characters)

* -6 - -4. The Whacks of Frostfield (12/31/2013)
* -3 - -2. The Whacks of Powder Point (3/20/2015)
* -1b. James's Bad Day (7/28/2015)
* -1a. If there were only parmesan strips... (8/23/2015)
* 0. Pilot (9/7/2015)

## Volume 1 (11/14/2015-3/26/2016)

* 1a. (11/14/2015)
* 1b. (11/21/2015)
* 1c. (11/28/2015)
* 2a. (12/5/2015)
* 2b. (12/6/2015)
* 3a. (12/12/2015)
* 3b. (12/19/2015)
* 4. (12/24/2015)
* 5a. (1/2/2016)
* 5b. (1/9/2016)
* 5c. (1/16/2016)
* 6a. (1/23/2016)
* 6b. (1/30/2016)
* 7a. (2/6/2016)
* 7b. (2/13/2016)
* 8a. (2/20/2016)
* 8b. (2/21/2016)
* 9. (2/27/2016)
* 10a. (3/5/2016)
* 10b. (3/12/2016)
* 10c. (3/19/2016)
* 10d. (3/26/2016)

## Volume 2 (11/12/2016-4/8/2017)

* 11a. (11/12/2016)
* 11b. (11/19/2016)
* 12. (11/22/2016)
* 13-15. The Whacks of Whiskview (11/25/2016)
* 16a. 2147483647th Customer (11/26/2016)
* 16b. (12/3/2016)
* 16c. (12/10/2016)
* 16d. The Escape Room (12/17/2016)
* 17a. (12/31/2016)
* 17b. (1/7/2017)
* 18a. (1/14/2017)
* 18b. (1/21/2017)
* 19. (1/28/2017)
* 20a. (2/4/2017)
* 20b. Love on all levels (2/11/2017)
* 21a. (2/18/2017)
* 21b. The Girl That Got Grounded For Life (2/25/2017)
* 21c. The Phone Call (3/4/2017)
* 21d. (3/11/2017)
* 21e. The Channel Changer (3/12/2017)
* 22a. (3/11/2017)
* 22b. (3/18/2017)
* 23a. The Red Bench 1 (3/20/2017)
* 23b. The Red Bench 2 (3/21/2017)
* 23c. The Red Bench 3 (3/22/2017)
* 23d. The Red Bench 4 (3/23/2017)
* 23e. The Red Bench 5 (3/24/2017)
* 24a. (4/1/2017)
* 24b. Daphne's Sweet Job! (4/8/2017)

## Volume 3 (11/11/2017-3/24/2018)

* 25. Sleepover at Forest's (11/11/2017)
* 26a. Massive Colorful Sliming (11/17/2017)
* 26b. Logan's Birthday (11/24/2017)
* 27. (12/2/2017)
* 28a. (12/9/2017)
* 28b. (12/16/2017)
* 29a. (12/23/2017)
* 29b. (12/30/2017)
* 30. Street Sweets (1/5/2018)

## Volume 4 (11/17/2018-2019)

31. Waterfield in Danger (11/17/2018)

32. Scooperia Party!
Tableau Simplex Method

## **Problem**

Consider the LP problem:

M
a
x

2
x
−
3
y
z

x
y
z
≤
10

4
x
−
3
y
z
≤
3

2
x
y
−
z
≤
10

x
,
y
,
z
≥
0

{\displaystyle {\begin{aligned}Max\quad 2x-3y+z\\x+y+z\leq 10\\4x-3y+z\leq 3\\2x+y-z\leq 10\\x,y,z\geq 0\end{aligned}}}

Construct the initial tableau T1 and solve the tableau using the simplex method.

## **Solution**

**Constructing the tableau**

Introduce slack variables s1, s2, s3.

**Pivot 1**

To identify the entering variable, look at the row in the bottom (row 4 in this problem). Look for the most negative entry. This is, -2, found under column 1. Hence the entering variable is **x**. To select the entering variable, look the numbers on the far right and perform division calculations with respect to the entries under column 1.

10

/

# 1
10

{\displaystyle 10/1=10}

3

/

# 4
0.75

{\displaystyle 3/4 = 0.75}

10

/

# 2
5

{\displaystyle 10/2=5}

Select the row with the smallest quotient (row 2). Hence the leaving variable is **s2**.

Now clear the other coefficients in column 1 using row-reduction methods. For this pivot, perform the following computations:

4

R

1

−

R

2

{\displaystyle 4R\_{1}-R\_{2}}

2

R

3

−

R

2

{\displaystyle 2R\_{3}-R\_{2}}

2

R

4

R

2

{\displaystyle 2R\_{4}+R\_{2}}

The tableau becomes

**Pivot 2**

In tableau 2, column 3 of row 4 has the most negative entry, so **z** is the entering variable.

37

/

# 3
12.333333

{\displaystyle 37/3=12.333333}

3

/

# 1
3

{\displaystyle 3/1=3}

Omit

17

/

−
3

{\displaystyle 17/-3}

 because non-positive denominators are ignored.
Since row 2 has the smallest quotient, **x** is the leaving variable.

Now clear the other coefficients in column 3 using row-reduction methods. For this pivot, perform the following computations:

R

1

−
3

R

2

{\displaystyle R\_{1}-3R\_{2}}

R
3

3

R
2

{\displaystyle R\_3+3R\_2}

R

4

R

2

{\displaystyle R\_{4}+R\_{2}}

There are no more negative entries in row 4! This means we are done! The maximum value can be calculated in to ways.

**Method 1:** Look at row 4 and the rightmost entry, then divide by the entry to its immediate left.

# Z
6

/

# 2
3

{\displaystyle Z=6/2=3}

**Method 2:** Look at the variables in the rightmost column and look at the values associated with non-slack variables. For this tableau, we only have

# z
3

{\displaystyle z=3}

. This means

(
x
,
y
,
z
# )
(
0
,
0
,
3
)

{\displaystyle (x,y,z)=(0,0,3)}

. Substitutes these values into the original objective function.

# Z
2
(
0
)
−
3
(
0
)
# 3
3

{\displaystyle Z=2(0)-3(0)+3=3}
Surface Area of a Sphere

## **Problem**

Derive the formula for the surface area of a sphere given its diameter.
Hint: the formula for the volume of a sphere is

π

d
3

6

{\displaystyle \frac{\pi d^3}{6}}

.

## **Solution**

Let

V

{\displaystyle V}

 be the volume of a sphere, and

S

{\displaystyle S}

 be the surface area of a sphere. The volume of the sphere can be approximated by many square pyramids (in truth it could be cones or pyramids with different polygon bases). The height of each pyramid is the radius of the sphere

r

{\displaystyle r}

, and base area

A

{\displaystyle A}

. Hence, the volume of the sphere is approximately

V
≈

r
3

(
n
A
)

{\displaystyle V\approx\frac{r}{3} (nA)}

, where

n
A
→
S

{\displaystyle nA \rightarrow S}

 as

n
→
∞

{\displaystyle n \rightarrow \infty}

.

As the sphere is subdivided into more, vanishingly small pyramids, we get the relation

# V

r
3

# S

d
6

S

{\displaystyle V=\frac{r}{3} S=\frac{d}{6} S}

.

Since the volume of the sphere is

# V

π

d
3

6

{\displaystyle V = \frac{\pi d^3}{6} }

, we discover that

# S
π

d
2

{\displaystyle S=\pi d^2 }

.
Sarah

Full name: Sarah Lexi Jackson (Sarah L. Jackson)

Born: April 12

Age: 18

After watching "100 Things To Do Before High School" at the age of 13, she decided to do those things over the last few months of her last year of middle school to prepare for high school.

## Facts about her

* She likes Mission figs.
* She is allergic to peanuts.
* She owns over 320 pairs of flip flops, in various colors and designs, and stores them in a closet.
* She has a dog named Orangey, but she usually calls it Lemonade due to it    being more yellow than orange. The dog was given to her by her parents on her 10th birthday, and is now currently 7.
* She never buys lunch in school; she always brings just a fruit and/or candy, and a drink.
* She is a fan of rainbow sherbet, tutti fruitti, and other fruity things.
* Her favorite fruits are berries (especially blueberries) followed by apples and oranges.
* She can count to 10 in Chinese.
* She is good at checkers.
* Her favorite chipmunk is Simon.
* Her favorite Loud House character is Lori.
* Her favorite character that associates with The Casagrandes is Bobby.
* Once she graduates, she plans to go to a college in 324ent world.

## Additional info

* Height: 5'3
* Zodiac sign: Aries
* Favorite color: Blue
* Handedness: Right
* Belly button: Innie
* Ticklish level (1-10): 9
* Shoe size: 8
* Toe color: Blue (changes over time since she loves showing them off)
* Food day she was born on: Grilled Cheese Sandwich Day
* Phone number: (193) 931-7942
Spectral Theory and the Inertia Tensor

## **Problem**

The general moment of inertia tensor

I

α
β

{\displaystyle { I }\_{ \alpha \beta } }

 is represented by a

3
×
3

{\displaystyle 3 \times 3
}

 real-symmetric matrix. Show that there exist an orthogonal matrix

P

{\displaystyle P}

 and diagonal matrix

D

{\displaystyle D}

 such that

# D

P

T

I

α
β

P

{\displaystyle D = { P }^{ T }{ I }\_{ \alpha \beta }P }

. Furthermore, deduce that

I

α
β

=
P
D

P

T

{\displaystyle { I }\_{ \alpha \beta } = PD{ P }^{ T } }

.

## **Solution**

We first prove a general property in spectral theory: If

A

{\displaystyle A}

 is a Hermitian matrix, there exists a unitary matrix \(P\), the conjugate transpose of the unitary matrix

P

∗

{\displaystyle {P}^{\*} }

 and a diagonal matrix

D

{\displaystyle D}

 such that

P

∗

A
# P
D

{\displaystyle {P}^{\*}AP =D }

.

We begin by finding

P

{\displaystyle P}

 such that

P

∗

A
# P
U

{\displaystyle {P}^{\*}AP = U }

, where

U

{\displaystyle U}

 is an upper triangular matrix. Since Hermitian matrices follow the property

A

∗

=
A

{\displaystyle {A}^{\*} = A }

, we can deduce the following:

(

P

∗

A
P

)

∗

=

U

∗

{\displaystyle {\left({P}^{\*}AP\right)}^{\*} = {U}^{\*} }

P

∗

A

∗

# P

U

∗

{\displaystyle {P}^{\*}{A}^{\*}P = {U}^{\*} }

P

∗

A
# P

U

∗

{\displaystyle {P}^{\*}AP = {U}^{\*} }

The conjugate transpose of an upper triangular matrix is a lower triangular matrix. Let

L

{\displaystyle L}

 be the lower triangular matrix. From the observation that

# U
L

{\displaystyle U = L }

 implies

P

∗

A
# P
D

{\displaystyle {P}^{\*}AP =D }

.

However, we know that the inertia tensor is real-symmetric. Hence, it follows the property: If

A

{\displaystyle A}

 is a real-symmetric matrix, there exists an orthogonal matrix

P

{\displaystyle P}

, the transpose of the orthogonal matrix

P

T

{\displaystyle {P}^{T} }

 and a diagonal matrix \(D\) such that

P

T

A
# P
D

{\displaystyle {P}^{T}AP =D }

.

Since real-symmetric matrices satisfy the property

A

T

=

A

{\displaystyle {A}^{T} = {A} }

 and orthogonal matrices satisfy the property

P

T

=

P

−
1

{\displaystyle {P}^{T} = {P}^{-1} }

, the proof is structurally identical to the one above.

Therefore, it follows that

P

T

I

α
β

# P
D

{\displaystyle { P }^{ T }{ I }\_{ \alpha \beta }P = D }

and

I

α
β

=
P
D

P

T

{\displaystyle { I }\_{ \alpha \beta } = PD{ P }^{ T } }

where

P

{\displaystyle P}

 is the rotation matrix, and

D

{\displaystyle D}

 is a diagonal matrix containing the principle axes of rotation.
Projectile Motion

## **Problem**

Consider a projectile fired at an initial velocity

v
i

{\displaystyle v\_i}

 at firing angle of

θ

{\displaystyle \theta}

 above the ground. Ignore air resistance or the effects of a rotational effects of the Earth.

**Part 1:** Calculate the time it takes for the projectile to land at the same altitude.

**Part 2:** Calculate the range of the projectile in terms of

v
i

{\displaystyle v\_i}

,

θ

{\displaystyle \theta}

, and

g

{\displaystyle g}

.

**Part 3:** Calculate the maximum altitude that the projectile is able to attain.

## **Solution**

This problem must be analyzed in two directions.

x
f

=

x
i

v

i
,
x

t

1
2

a
x

t
2

{\displaystyle x\_f=x\_i+{v}\_{i,x} t+ \frac{1}{2} a\_x t^2 }

y
f

=

y
i

v

i
,
y

t

1
2

a
y

t
2

{\displaystyle y\_f=y\_i+{v}\_{i,y} t+ \frac{1}{2} a\_y t^2 }

Treat the initial point

(

x
i

,

y
i

# )
(
0
,
0
)

{\displaystyle (x\_i,y\_i)=(0,0) }

. There is no acceleration in the x-direction, and the acceleration in the x-direction is

a
x

=
−
g

{\displaystyle a\_x=-g }

.

x
f

=

v
i

cos
⁡
(
θ
)
t

{\displaystyle x\_f=v\_i \cos (\theta) t }

y
f

=

v
i

sin
⁡
(
θ
)
t
−

1
2

g

t
2

{\displaystyle y\_f=v\_i \sin (\theta) t- \frac{1}{2} gt^2 }

**Part 1**

Since the projectile lands at the same altitude, the displacement in the y-direction is 0.

# 0

v
i

sin
⁡
(
θ
)
t
−

1
2

g

t
2

{\displaystyle 0=v\_i \sin (\theta) t- \frac{1}{2} gt^2 }

# 0
t

(

v
i

sin
⁡
(
θ
)
−

1
2

g
t

)

{\displaystyle 0=t \left(v\_i \sin (\theta) - \frac{1}{2} gt \right) }

This means the two times in which

y
f

=
0

{\displaystyle y\_f=0 }

 are

# t
0

{\displaystyle t=0}

 and

# t

2

v
i

sin
⁡
(
θ
)

g

{\displaystyle t=\frac{2v\_i \sin (\theta)}{g} }

. Of course the second time is the landing time.

**Part 2**

x
f

=

v
i

cos
⁡
(
θ
)
t

{\displaystyle x\_f=v\_i \cos (\theta) t }

x
f

=

v
i

cos
⁡
(
θ
)

(

2

v
i

sin
⁡
(
θ
)

g

)

{\displaystyle x\_f=v\_i \cos (\theta) \left( \frac{2v\_i \sin (\theta)}{g} \right) }

x
f

=

2

v

i

2

sin
⁡
(
θ
)
cos
⁡
(
θ
)

g

{\displaystyle x\_f= \frac{2{v}^{2}\_{i} \sin (\theta) \cos (\theta)}{g} }

.
Since

sin
⁡
(
2
θ
# )
2
sin
⁡
(
θ
)
cos
⁡
(
θ
)

{\displaystyle \sin (2\theta) = 2 \sin (\theta) \cos (\theta) }

, the range of the projectile is

x
f

=

v

i

2

sin
⁡
(
2
θ
)

g

{\displaystyle x\_f= \frac{{v}^{2}\_{i} \sin (2\theta)}{g} }

.
**Part 3**

The top of the trajectory occurs at half the time of landing, so

t

1

/

2

=

v
i

sin
⁡
(
θ
)

g

{\displaystyle {t}\_{1/2} = \frac{v\_i \sin (\theta)}{g} }

.
Substitute this time to the displacement equation in the y-direction.

y

m
a
x

=

v
i

sin
⁡
(
θ
)

t

1

/

2

−

1
2

g

(

t

1

/

2

)

2

{\displaystyle {y}\_{max} =v\_i \sin (\theta) {t}\_{1/2}- \frac{1}{2} g \left({t}\_{1/2} \right)^2 }

y

m
a
x

=

v
i

sin
⁡
(
θ
)

v
i

sin
⁡
(
θ
)

g

−

1
2

g

(

v
i

sin
⁡
(
θ
)

g

)

2

{\displaystyle {y}\_{max} =v\_i \sin (\theta) \frac{v\_i \sin (\theta)}{g} - \frac{1}{2} g\left(\frac{v\_i \sin (\theta)}{g}\right)^2 }

y

m
a
x

=

v

i

2

sin
2

⁡
(
θ
)

2
g

{\displaystyle {y}\_{max} =\frac{{v}^{2}\_{i} \sin^2 (\theta)}{2g} }

.
Variance and Expectation

## **Problem**

**Part 1:** Begin with the definition of the variance

V
a
r
(
X
# )
E
[
(
X
−
μ

)
2

]

{\displaystyle Var(X) = E[(X-\mu)^2] }

.
Using the linear property of expectation

E
[
α
X
β
Y
# ]
α
E
[
X
]
β
E
[
Y
]

{\displaystyle E[\alpha X + \beta Y] = \alpha E[X] + \beta E[Y] }

, show that

V
a
r
(
X
# )
E
[

X
2

]
−

μ

2

{\displaystyle Var(X) = E[X^2] - {\mu}^2 }

.

Note that

# μ
E
[
X
]

{\displaystyle \mu = E[X] }

.

**Part 2:** Prove

V
a
r
(
c
X
# )

c
2

V
a
r
(
X
)

{\displaystyle Var(cX) = c^2 Var(X) }

.

## **Solution**

**Part 1**

V
a
r
(
X
# )
E
[
(
X
−
μ

)
2

]

{\displaystyle Var(X) = E[(X-\mu)^2] }

V
a
r
(
X
# )
E
[

X
2

−
2
μ
X

μ

2

]

{\displaystyle Var(X) = E[X^2 - 2\mu X + {\mu}^2] }

V
a
r
(
X
# )
E
[

X
2

]
−
E
[
2
μ
X
]
E
[

μ

2

]

{\displaystyle Var(X) = E[X^2] - E[2\mu X] + E[{\mu}^2] }

V
a
r
(
X
# )
E
[

X
2

]
−
2
μ
E
[
X
]

μ

2

E
[
1
]

{\displaystyle Var(X) = E[X^2] - 2\mu E[X] + {\mu}^2 E[1] }

Since

# μ
E
[
X
]

{\displaystyle \mu = E[X] }

 and

E
[
1
# ]
1

{\displaystyle E[1] = 1 }

,

V
a
r
(
X
# )
E
[

X
2

]
−
2

μ

2

μ

2

{\displaystyle Var(X) = E[X^2] - 2{\mu}^2 + {\mu}^2 }

V
a
r
(
X
# )
E
[

X
2

]
−

μ

2

{\displaystyle Var(X) = E[X^2] - {\mu}^2 }

.
Alternatively the variance can be written as

V
a
r
(
X
# )
E
[

X
2

]
−

(
E
[
X
]
)

2

{\displaystyle Var(X) = E[X^2] - {(E[X])}^2 }

.

**Part 2**

From part 1,

V
a
r
(
X
# )
E
[

X
2

]
−

μ

2

=

{\displaystyle Var(X) = E[X^2] - {\mu}^2 = }

.
Since

# μ
E
[
X
]

{\displaystyle \mu = E[X] }

,

V
a
r
(
X
# )
E
[

X
2

]
−

(

E
[
X
]

)

2

{\displaystyle Var(X) = E[X^2] - \left(E[X] \right)^2 }

.
Subsequently,

V
a
r
(
c
X
# )
E
[
(
c
X

)
2

]
−

(

E
[
c
X
]

)

2

{\displaystyle Var(cX) = E[(cX)^2] - \left(E[cX] \right)^2 }

V
a
r
(
c
X
# )
E
[

c
2

X
2

]
−

(

c
E
[
X
]

)

2

{\displaystyle Var(cX) = E[c^2 X^2] - \left(c E[X] \right)^2 }

V
a
r
(
c
X
# )
E
[

c
2

X
2

]
−

c
2

(

E
[
X
]

)

2

{\displaystyle Var(cX) = E[c^2 X^2] - c^2 \left(E[X] \right)^2 }

V
a
r
(
c
X
# )

c
2

E
[

X
2

]
−

c
2

(

E
[
X
]

)

2

{\displaystyle Var(cX) = c^2 E[X^2] - c^2 \left(E[X] \right)^2 }

V
a
r
(
c
X
# )

c
2

(

E
[

X
2

]
−

(

E
[
X
]

)

2

)

{\displaystyle Var(cX) = c^2 \left(E[X^2] - \left(E[X] \right)^2 \right) }

V
a
r
(
c
X
# )

c
2

V
a
r
(
X
)

{\displaystyle Var(cX) = c^2 Var(X) }

.
Sunzi Suanjing Lower Scroll 33

## **Problem**

答曰：九萬匝。

術曰：置九百里，以三百步乘之，得二十七萬步。又以六尺乘之，得一百六十二萬尺。以車輪一丈八尺為法。除之，即得。

Answer: 90,000 revolutions.

Method: Set 900 *li,* multiply by 300 *bu* to obtain 270,000 *bu*. Then multiply by 6 *chi* to obtain 1,620,000 *chi*. Let [the circumference of ] the wheel, 1 *zhang* 8 *chi*, be the divisor. Divide to obtain the answer.

**Note:** 1 *li* = 300 *bu*, 1 *bu* = 6 *chi*, 1 *zhang* = 10 *chi*

## **Solution**

This problem is an exercise of unit conversions. The first step is to convert the distance between Luoyang and Chang'an from the unit *li* to *chi*.

900 *li* converts to 270,000 *bu*.

900

l
i
×

300

b
u

1

l
i

=
270
,
000

b
u

{\displaystyle 900 \; li \times \frac{300 \; bu}{ 1 \; li} = 270,000 \; bu }

270,000 *bu* converts to 1,620,000 *chi*.

270
,
000

b
u
×

6

c
h
i

1

b
u

=
1
,
620
,
000

b
u

{\displaystyle 270,000 \; bu \times \frac{6 \; chi}{1 \; bu} = 1,620,000 \; bu }

Since one rotation of the wheel is 18 *chi*, divide the distance between Luoyang and Chang'an in chi by 18 *chi*.

1
,
620
,
000

c
h
i

18

c
h
i

=
90
,
000

{\displaystyle \frac{1,620,000 \; chi}{18 \; chi} = 90,000 }

Thus, the wheel makes 90,000 rotations from Luoyang to Chang'an.
Logan

Full name: Logan Peters

Born: January 27, 1993 (age 28)

No occupation yet, but is a fan of bright colors and polka dots. A fan of carnival foods, boat trips, and horses. She was the wife of Lime, who was killed in a gas explosion as of July 17, 2018.

## Additional info

* Occupation: Unknown
* Height: 5'10
* Zodiac sign: Aquarius
* Favorite colors: Bright blue, neon green, teal, purple, pretty much any bright/neon color
* Handedness: Right
* Belly button: Innie
* Favorite foods: Freeze pops (especially the pink ones!), chocolate cake
* Ticklish level (1-10): 11.5
* Shoe size: 9
* Toe color: Blue
* Food day she was born on: Chocolate Cake Day (she sure does love it!)
* Phone number: (297) 973-9474

## Fun facts

* Her username on social media is LoganPeters127.
* She only prefers flip flops and sandals, and thinks other shoes are boring.
* She won The Voice Waterland (a talent show) in 2015.
* She placed second in the 2014 Flipversevision Song Contest with her song "Luau Punch".
* She is an Aquarius, and her favorite colors go well with her sign.
* Her favorite character that associates with The Casagrandes is Maria.
Intensity Levels

In 1033Forest's TTS videos, an intensity level is used to determine how crazy a scene is.

## The levels

* Level 0 (blue): Nothing crazy happening, everything is normal (no music or a stock track)
* Level 1 (green): Things are getting a little crazy (for example, Microsoft Sam just got his job) (no music or a stock track)
* Level 2 (yellow): A bit stronger than level 1 (for example, after his boss tells Sam he hasn't done his job correctly) (stock track) (25% chance of test card)
* Level 3 (orange): Things are getting crazy here (for example, Sam eating all the tacos on taco night) (Batty McFaddin) (50% chance of test card)
* Level 4 (red): Things get REALLY intense (for example, major technical difficulties, or if a robber or NSA officer or other person with weapons comes in) (this track) (100% chance of test card; a test card always shows up at this point due to possible violence or other stuff)
Stacey

Born: April 13, 1997 (age 23)

More info coming soon

## Facts about her

* Both of her parents were born on the same day (March 11), but on different years (her mother was born 1977 and her father 1980). When Stacey was 11, they both died, and her brother Brai, who was 20 at the time, had to take care of her (he is 32 now)
* Her mother died at 29, and her father at 26.
Movie Theater

The **Movie Theater** is also unnamed, but is a place in the 312 Entertainment world.

In here, customers go there to watch a movie. It plays some real movies, and some fictional ones produced by 312 Entertainment. There are 312 theaters there.

## Floors

* C. Concession Stand
* 1. Theaters 1-100
* 2. Theaters 101-200
* 3. Theaters 201-300
* 4. Theaters 301-312

## Concession Stand Items

#### Popcorn (each one has a different color box)

* Buttered (yellow box)
* Caramel (blue box)
* Kettle (green box)
* Chocolate (brown box)
* Red Hot (red box)
* Cinnamon (purple box)
* Cheddar (orange box)
* Rainbow candy coated (teal box)

#### Drinks (needs better names)

* Fizzy Fizz (also available in clear)
* Diet Fizzy Fizz
* Jittery Green
* Dr. 312
* Twisted Lemon
* Orange POP! (also in red, yellow, purple, blue, green, and teal)
* Root Beer
Measurement of the Cycloid

## **Problem**

A **cycloid** is the curve traced by a point on a circle as it rolls along a straight line. The standard cycloid is usually expressed in parametric form:

{

# x
r
(
t
−
sin
⁡
t
)

# y
r
(
1
−
cos
⁡
t
)

{\displaystyle
\begin{cases}
x = r(t - \sin t) \\
y = r(1 - \cos t)
\end{cases}
}

where

0
≤
t
≤
2
π

{\displaystyle 0 \leq t \leq 2\pi }

 and

r

{\displaystyle r}

 is the radius of the rolling circle that traces out the cycloid.

**Part 1:** Determine the area under the cycloid. Compare the result to the area of the rolling circle.

**Part 2:** Determine the arclength of the cylcoid.

## **Hint**

(1) Use the integral

∫

cos
2

⁡
x

d
# x

x
2

1
4

sin
⁡
2
x

{\displaystyle \int \cos^2 x \; dx = \frac{x}{2} + \frac{1}{4}\sin 2x }

.

## **Solution**

**Part 1**

Set up and evaluate the integral for calculating the area under the curve.

A

=

∫

0

2
π
r

y

d
x

=

∫

0

2
π

r
2

(
1
−
cos
⁡
t

)
2

d
t

=

r
2

∫

0

2
π

1
−
2
cos
⁡
t

cos
2

⁡
t

d
t

=

r
2

[

t
−
2
sin
⁡
t

t
2

1
4

sin
⁡
2
t

0

2
π

]

=
3
π

r
2

.

{\displaystyle
\begin{align}
A &= \int\_{0}^{2 \pi r} y \; dx \\[5 pt]
&= \int\_{0}^{2 \pi} r^2(1 - \cos t)^2 \; dt \\[5 pt]
&= r^2\int\_{0}^{2 \pi} 1 - 2\cos t + \cos^2 t \; dt \\[5 pt]
&= r^2 \left[ t - 2\sin t + \frac{t}{2} + \frac{1}{4}\sin 2t \Biggr|\_{0}^{2\pi} \right] \\[5 pt]
&= 3 \pi r^2.
\end{align}
}

The area under the cycloid is three times the area of the rolling circle.

**Part 2**

Set up and evaluate the integral for calculating arclength.

L

=

∫
0

2
π

(

d
x

d
t

)

2

(

d
y

d
t

)

2

d
t

=

∫
0

2
π

(
r
(
1
−
cos
⁡
t
)

)
2

(
r
sin
⁡
t

)
2

d
t

=

∫
0

2
π

r

2
−
2
cos
⁡
t

d
t

=
2
r

∫
0

2
π

sin
⁡

t
2

d
t

=
−
4
r

[

cos
⁡

t
2

0

2
π

]

=
8
r
.

{\displaystyle
\begin{align}
L &= \int\_0^{2\pi} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \; dt \\[5 pt]
&= \int\_0^{2\pi} \sqrt{(r(1 - \cos t))^2 + (r\sin t)^2} \; dt \\[5 pt]
&= \int\_0^{2\pi} r \sqrt{2 - 2\cos t} \; dt \\[5 pt]
&= 2r\int\_0^{2\pi} \sin \frac{t}{2} \; dt \\[5 pt]
&= -4r\left[\cos \frac{t}{2} \Biggr|\_{0}^{2\pi} \right] \\[5 pt]
&= 8r.
\end{align}
}
Lotion Paint Colors

This is the list of colors and scents for Grace's line of lotion "paints." Grace created this line to combine softening the skin, with the fun and mess of art. The colors disappear right after they're rubbed into skin.

## Color and Scent List

* Red (Cherry)
* Orange (Orange)
* Golden Yellow (Butterscotch)
* Yellow (Banana)
* Lime Green (Strawberry Kiwi)
* Green (Apple)
* Light Green (Mint)
* Teal (Fruit Punch)
* Light Blue (Blue Raspberry)
* Blue (Blueberry)
* Light Purple (Cotton Candy)
* Purple (Grape)
* Light Pink (Watermelon)
* Pink (Strawberry)
* Brown (Chocolate)
* White (Vanilla)
* Black (Licorice)
324 Entertainment Awards

The 324 Entertainment Awards are an award ceremony hosted every year on a Saturday night in March-May. This award show honors the year's biggest acts done in the 324 Entertainment world, as well as some other, non-324 Entertainment related categories, including favorite software programs. The winners receive a black clipboard figure with a "324" on it as the trophy (it's a clipboard because that's the old logo from 2017-2018).

## Dates for future ceremonies

* 2018 ceremony: April 7, 2018
* 2019 ceremony: March 30, 2019
* 2020 ceremony: April 11, 2020
Stair Cave

A new building to be introduced to the 324 Ent. world underground which the main attraction is 324 floors of stairs. If you want to face the biggest challenge yet, then go for this building and see how many stairs can you climb. Once you reach the top, you can slide your way down the building into a pool of foam blocks, which is part of the underground cave in a hidden room, or go to the place up at the surface which currently isn't known yet.

Emergency slides exist after every certain amount of floors if you want to get off early because of a certain condition or something.

Each floor will have a random color chosen from 27 random colors. The first floor is black, and the 324th floor is white, black and white will be randomly used 11 times throughout the building, while the other 25 colors will be randomly used 12 times.

The first floor of this stairs challenge is located 0.61 miles below the surface and where the 324th floor leads to is yet to be known. The stairs challenge is part of an underground cave/building which also includes an elevator and tubes to go up/down.
Love on all levels

Valentine's Day special 2017. Inspired by this: https://www.youtube.com/watch?v=UKOD44WQDnA

A part of 324 Entertainment Shorts: Volume 2. DO NOT EDIT

Release date: February 11, 2017

Rated: 12

## Cast

* 1033Forest (Director/filmer)
* Gigi
* Carey
* Clara
* Claudia

## Transcript

* (1033Forest claps clapboard)
* Gigi: I never really saw myself as that kind of person.
* Clara: She'd been asking me for weeks, sending invite after invite.
* Carey: Hmm, 3 years. Wow... I can't believe it's been 3 years.
* Clara: Until I thought, you know, I should just go for it.
* Claudia: But then, Gigi just kept pushing me. She'd say...
* Gigi: "Go on! Give it a try!"
* Clara: I know that sounds a bit cliche, but we just had this...amazing connection.
* Carey: Love at first sight? I don't know. I mean we kept seeing each other on breaks, at lunch...
* Clara: Sure, sometimes it's been hard and sometimes I even felt like giving up.
* Claudia: And now I feel so liberated!
* Clara: I'm on level 1033!
* Carey: Best. Game. Ever.
* Clara: It's love. Love on all levels.
* (Candy Crush logo appears)
* 1033Forest: (in background) Sugar Crush!
Cotton Candy Flavors

This is a list of flavors of cotton candy sold at 432 Carnival.

## Flavors

* Cherry Berry (Cherry) (Red)
* O-Jay (Orange) (Orange)
* Banana Bonanza (Banana) (Yellow)
* Leapin' Lime (Lime) (Green)
* Boo Blue (Blue Raspberry) (Blue)
* Spookie Fruiti (Grape) (Purple)
* Silly Nilly (Vanilla) (Pink)
* Birthday Cake (White)
* Maple (Brown)

More to be added soon
How to make blueberry swizzles

* 1 batch white icing
* Blue food coloring
* Blueberry flavoring
* Empty rolled wafers
* Blue edible marker

1. Mix the icing with blue food coloring and blueberry flavoring to make the blueberry filling. Use a gel food coloring so your icing won't be thin and soupy.
2. Make the striped pattern on your rolled wafers with the edible marker. It's best to do this before filling them so you won't make a mess.
3. Put the blueberry filling in a piping bag then pipe your filling inside your decorated wafers.
4. Serve and eat within 1-2 days.
Laplace Transform

## **Problem**

Solve

y
″

−

y
′

−
2
# y
4

e

−
t

{\displaystyle y'' - y'- 2y = 4{e}^{-t} }

 subject to the initial-values

y
(
0
# )
0

{\displaystyle y(0) = 0 }

 and

y
′

(
0
# )
0

{\displaystyle y'(0) = 0 }

.

## **Solution**

The following Laplace transforms will be useful for this differential equation.

L
[
y
# ]
Y
(
s
)

{\displaystyle L[y] = Y(s) }

L
[

y
′

# ]
s
Y
(
s
)
−
⋅
y
(
0
)

{\displaystyle L[y'] = s Y(s) - \cdot y(0) }

L
[

y
″

# ]

s
2

Y
(
s
)
−
s
⋅
y
(
0
)
−

y
′

(
0
)

{\displaystyle L[y''] = s^2 Y(s) - s\cdot y(0) - y'(0) }

L
[

e

a
t

# ]

1

s
−
a

{\displaystyle L[{e}^{at}] = \frac{1}{s-a}}

This means

(

s
2

Y
(
s
)
−
s
⋅
0
−
0

)

−

(

s
Y
(
s
)
−
0

)

−
2
Y
(
s
# )

4

s
1

{\displaystyle \left(s^2 Y(s) - s\cdot 0 - 0\right) - \left(s Y(s) - 0\right) -2 Y(s) = \frac{4}{s+1} }

Y
(
s
)

(

s
2

−
s
−
2

)

=

4

s
1

{\displaystyle Y(s) \left(s^2 -s -2 \right) = \frac{4}{s+1} }

Y
(
s
# )

4

(
s
−
2
)

(
s
1
)

2

{\displaystyle Y(s) = \frac{4}{(s-2){(s+1)}^{2}} }

By partial fractions decomposition,

Y
(
s
# )

4

/

9

s
−
2

−
4

/

9

s
1

−
4

/

3

(
s
1
)

2

{\displaystyle Y(s) = \frac{4/9}{s-2} + \frac{-4/9}{s+1} + \frac{-4/3}{{(s+1)}^{2}} }

.
The inverse Laplace transform is

L

−
1

[
Y
(
s
)
# ]

4
9

e

2
t

−

4
9

e

−
t

−

4
3

t

e

−
t

{\displaystyle {L}^{-1} [Y(s)] = \frac{4}{9} {e}^{2t} - \frac{4}{9} {e}^{-t} - \frac{4}{3} t {e}^{-t} }

Therefore the solution to this initial-value problem (IVP) is

y
(
t
# )

4
9

e

2
t

−

4
9

e

−
t

−

4
3

t

e

−
t

{\displaystyle y(t) = \frac{4}{9} {e}^{2t} - \frac{4}{9} {e}^{-t} - \frac{4}{3} t {e}^{-t}}

.
The Red Bench 1

The first of 5 episodes released on Red Bench Week, this week will end the second volume of 312 Entertainment Shorts.

Release date: March 20, 2017

DO NOT EDIT

## Cast

* Jarrod
* Ciyana
* Daphne
* Lillian
* Clara
* 1033Forest

## Transcript

* (Jarrod sits on a red bench, Ciyana approaches)
* Ciyana: Whatcha doing?
* Jarrod: Sitting on a red bench.
* Ciyana: Can I join you?
* Jarrod: Sure.
* (Ciyana sits. Daphne approaches)
* Daphne: Whatcha doing?
* Both: Sitting on this red bench.
* Daphne: Can I join?
* Both: Sure.
* (Daphne sits. Lillian approaches)
* Lillian: Whatcha doing?
* All 3: Sitting on this red bench.
* Lillian: Can I join you?
* All 3: Sure!
* (Lillian sits. Clara approaches.
* Clara: Whatcha doing?
* All 4: Sitting on this red bench.
* Clara: Can I join?
* All 4: Sure!
* (Clara sits. 1033Forest approaches)
* 1033Forest: Whatcha doing?
* All 5: Sitting on this red bench.
* 1033Forest: You mean the one I moved over there? (points in opposite direction)
* (all 5 fall off the bench)

## Trivia

* The Red Bench parts 2, 3, and 4 don't need separate pages for now, they are almost the same except another character is added and the bench is just painted/glued together/something different/etc.
Crazy 8 Vine

A very rare vine that appears in the vines screensaver. Its appearance is a small, thin 8 with no leaves on it, 2 rings on top of each other.

There is only one of this vine in 432 Carnival's Jungle of Rainbow Vines. If it breaks, it will need to be replaced and the person who broke it will have to pay a $888.88 fine.

## Trivia

* This vine is named after the card game Crazy Eights .
Good times leo lion = king of african hybrds

[ based on lion king 1 n 2 n prince n pauper [ novel ] ]

## animals appers

asian elephants

giant pandas

bactrain camels

bengal tigers

black rhinos

panthers

komodos

tapirs

musk oxen

african elephants

quing pandas

dromdary camels

sumatra tigers

javen rhinos

lions

giraffes

hippos

zebras

wild beest

warthogs

hyenas

gazelles

cars

buses

bikes

planes

diggers

vans

boats

safari parks

jungles

towns

farms

woodlands

beaches

## cast

leo lion

tooey lion cub

asian n african elephants

black n javen rhinos

mountain n lowland gorillas

bengal n sumatran tigers

lions

giraffes

hippos [ aka elephants ]

zebras

bactrain n domdary camels

bears

leopards jaguars n cheetahs

mooses

dinos

horses

oxen

sheep

pigs

humans

donkeys

goats

dogs n cats

[ out of animals n humans FROM leo lion JUNGLE BOOK CINDERELLA SNOW WHITE HERCULES ALICE IN WONDERLAND JUNGLE KING GOLDEN NOAH ‘S ARK PRINCE AND PAUPER golden prince n pauper Son of the lion king Lucy n lionel jungle kingdom [ seaside squad ] rocky railway [ tv show ] panda mania [ tv show ] imagin me = tarzan of the apes [ 2020 story app ] cool school classics = cool tales of 3 bears [ 2 2020 cool school films ] dot n the koala [ 2020 tv show ] jung jung jungle book 1 n 2 [ 2020 films ]  Rock land book ]

[ all appers in leo lion [ main ] JUNGLE BOOK CINDERELLA SNOW WHITE HERCULES ALICE IN WONDERLAND JUNGLE KING GOLDEN NOAH ‘S ARK PRINCE AND PAUPER golden prince n pauper Son of the lion king Lucy n lionel jungle kingdom [ seaside squad ] rocky railway [ tv show ] panda mania [ tv show ] imagin me = tarzan of the apes [ 2020 story app ] cool school classics = cool tales of 3 bears [ 2 2020 cool school films ] dot n the koala [ 2020 tv show ] jung jung jungle book 1 n 2 [ 2020 films ]  Rock land book clips ]

[ rorkrys of hildenbough asian african elephant bengal sumatran tiger monkey penguin wild boy wild girl savannah ]
Lime

Real full name: Ollie F. Alfonso

Born: June 2, 1988

Died: July 17, 2018 (aged 30)

Lime is a customer who got his nickname because of his love of limes. He is Logan's husband. On July 17, 2018, he was killed in a gas explosion, and Logan and the rest of the 324 Entertainment community will all remember him.
Mr. Toonster

**Mr. Toonster** is 1033Forest's YouTube channel. He has owned the channel since 2013, but didn't come up with the name Mr. Toonster until

   2015. On his channel he has posted the Windows 3.1 Errors series, and his experiements on computer programs, operating systems, and other obscure software. He even planned to make more Microsoft Sam and other TTS voice videos back in 2016, but he never got around to doing those.

When the channel rebranded as Mr. Toonster in 2015, he was known as "a guy born in c. 2000 but is never on until 3:30", beacuse of him always coming back home from his high school at 3:30 at the time.

He is also known for testing higher quants on the DDR and PIU games in two of his videos.

And a warning: Very few of his videos have questionable content in them, but most of his videos are clean.

## Mr. Toonster Productions

In 2017, Mr. Toonster founded Mr. Toonster Productions, the company named after him that has produced the previous Windows 3.1 Errors series, the "Has BSOD" video series, and The Ultimate BSOD Compilation films. In 2019, Sdq2002Productions has acquired 50% of the company for $15,000,000 and has became one of the best BSOD mergers in the world.
Casino

The **Casino** doesn't have a name yet, but it's a place located in the 311 Entertainment world. It is a family casino, which means, people under 21 are welcome there too. Unlike most casinos, this one has no alcohol, and the games there aren't gambling games. Kids are even welcome to visit; there are a few arcade games there that are especially for kids. It's like a grown up Chuck E Cheeses.

## Rules

1. All people under 21 will be asked to leave after 11 PM to ensure safety.
2. No bringing alcohol allowed, since this is a family casino
3. No smoking, for the same reason above

## Hours

* Sunday: 10 AM - 9 PM
* Monday-Thursday: 10 AM - 11 PM
* Friday-Saturday: 9 AM - 2 AM
* Closed on holidays: New Year, Easter, Thanksgiving, Christmas. Other holidays have reduced hours
1033Forest

Born: May 15, 2000 (age 23)

**1033Forest** Is a Discord and Wikia user who is a fan of mostly Nickelodeon shows. He particularly loves SpongeBob and The Loud House. He is the founder of the fictional company, 324 entertainment.

## Trivia

* 1033Forest's favorite food is pizza.
	+ His least favorite are certain vegetables.
* 1033Forest's favorite sodas are Pepsi, Sierra Mist, and Mountain Dew (his favorite Mountain Dew is Baja Blast or Voltage)
* 1033Forest has loved The Loud House since during the third season.
* 1033Forest loves doing roleplay.
* He first used the 1033Forest name back in 2015 on the Nick.com message boards, which are no longer available as of March 31, 2016.
* 1033Forest is exactly 2 weeks younger than Jack. (Harvey Beaks Number 1 Fan in Wikia)
* 1033Forest has a color rating scale used to rate episodes of cartoons and other Nick shows (preferably SpongeBob and The Loud House) whether they are great or terrible.
* 1033Forest sometimes has weird, random thoughts and interests that pop up in his head. Here's the list of them.
* 1033Forest’s favorite background characters of The Loud House are Becky and Cookie.
Loud House Day

Loud House Day is a day celebrated in the 324 Entertainment world every December 16. This day was chosen because in the show The Loud House, the address number of the Louds' residence is 1216.

You can celebrate this holiday in the real world, too, by dressing up as your favorite Loud House character and/or acting like them for this day.
Carey

Full name: Carey Smithy

Born: April 15, 1995 (age 27)

Her occupation is currently unknown. Based on the colors she wears, likes being artsy, and likes quite a few different holidays.

## Additional info

* Occupation: Unknown
* Height: 5'9
* Zodiac sign: Aries
* Favorite colors: Red, pink, purple, and blue
* Handedness: Right
* Belly button: Innie
* Favorite food: Strawberries, cherries, mostly red things
* Ticklish level (1-10): 10
* Shoe size: 8.5
* Toe color: Red
* Phone number: (107) 253-8759
The Flavor House

Plot: Nora and her friends get ready for the flavor play, which will be held in the Royal Woods theater.

Synopsis: We open up to Nora's house as she excitedly tells Zoey and Chase about the Flavor city play. Involving her and her friends protecting the city from evil candy. She excitedly calls Mimi, who is shaving her legs on her bathroom sink. Nora excitedly mentions how great the play will be and mentions how she and Mimi can hand out invitations. Mimi agrees, but loses her balance when Nora excitedly screams and falls out of the bathroom.

Nora and Mimi are then seen handing out flyers to their friends, who all agree to come see the play live. (Except Alex and Bud, who seem distracted by the game they're playing) Everyone takes their seats as Nora and her friends get ready for the play. With Stella and Kelsey trying to record the event. Nora, Mimi, Mitzie, and Kacy play the heroes, along with Shay, who is one of the police guards for candy jail. With Lucy playing the evil black licorice.

As the play goes on, Rosemary rushes in and is dismayed to see all the seats taken. Stella and Kelsey offer to move a few seats, so Rosemary can see Mimi perform. Though when trying to find a new seat, Stella ends up being startled by Haiku at the end of the row. Causing her to jump and land in Kelsey's arms. Resulting in them tripping backwards and falling into an isle.

The play ends with Lucy being arrested by Shay and being sent to candy jail as the audience cheers. As the play comes to a close, Zoey congratulates the girls, while Annie comes to hand out to pie as a way to celebrate. But she accidently trips over a prop. Causing her to fall to the floor, and toss the pie in Zoey's face. Who happily licks the delicious pie.
Jarrod

Born: November 11, 1989 (age 32)

A guy who is a regular in most restaurants in the series. His occupation is currently unknown. His favorite food is pasta, and his favorite pasta sauce is 3 cheese sauce (you can tell by the "3" on his shirt). His favorite color is orange (orders it a lot), and his favorite holiday is Halloween. His favorite types of pasta are macaroni, penne, and cellentani, pretty much anything tube-like.

## Additional info

* Occupation: Unknown
* Height: 5'8
* Zodiac sign: Scorpio
* Favorite color: Orange
* Handedness: Right
* Belly button: Outie
* Favorite food: Pasta and cheese
* Ticklish level (1-10): 2
* Shoe size: 9.5
* Food day he was born on: National Sundae Day
* Phone number: (086) 491-6398

## Trivia

* His birthday is on Veteran's day, or the day where all 4 digits of the date are same.
* Jarrod was the judge for Buttertown's televoting in the Flipversevision Song Contest 2014, who gave his 12 points to Logan.
Regedit Punch

* 1 gallon water
* 2 cups sugar
* ½ oz raspberry extract
* ½ oz vanilla extract
* Enough green and blue food coloring to make a dark teal color
* 1-2 lbs dry ice

## Instructions

1. Place gallon of water in large enough pitcher, then mix in the sugar, extracts, and food coloring.
2. Add the dry ice and wait until all the dry ice disappears/is vaporized.
3. Pour into a drink dispenser. Any slush formed can help cool down the punch. If you want to store it, put the punch in a gallon jug or in empty 2 liter bottles.