5Let’s learn the basics
Problem

Using linear functions

Using linear functions ⑷ 〜Filling and draining water tanks〜

4Figure 1 shows a cuboid-shaped tank that

Figure 1

Figure 2is 68 cm deep. Suppose this tank is filled withwater by using Pipes A and B together for the first
12 minutes, and then, after that, Pipe B only until
the tank is completely full. Figure 2 shows a graphof the relationship between x, the elapsed time inminutes measured from when the filling begins, andy, the depth of the water in the tank.
⑴

When 12≦ x ≦20, express y in terms of x.

⑵

How many minutes after filling began did the water depth become 53 cm?

⑶ If only Pipe A is used, how many cm per minute will the depth of the water increase?
Solution⑴

Find the expression of the line that passes through two points（12，48）and（20，68）.
Expressing it as y＝ax＋b,
since 48＝12a＋b and 68＝20a＋b, you get a＝2.5 and b＝18. Therefore, y＝2.5x＋18.

⑵

According to the graph, y＝53 when 12≦ x ≦20.
Substituting y＝53 in the expression you got in ⑴, you get 53＝2.5x＋18, x＝14.

⑶

According to the graph, the rate of increase is 48÷12＝4 cm per minute when you use both pipes.
Since it is 2.5 cm per minute when you use only Pipe B, it is 4−2.5＝1.5 cm when you use only
Pipe A.

Answer⑴

y＝2.5x＋18⑵

14 minutes after

9 Figure 1 shows is a cuboid-shaped tank that is⑶

1.5 cmFigure 1

Figure 2

43 cm deep. Suppose this tank is filled with waterby using only Pipe A for the first 12 minutes, andthen, after that, Pipes A and B together until thetank is completely full. Figure 2 shows a graph of the relationship between
x, the elapsed time in munutes measured from when the filling begins, andy, the depth of the water in the tank.
□⑴ When 10≦x≦18, express y in terms of x.

□⑵ How many minutes after filling began did the water depth become 36 cm?

□⑶ If only Pipe B is used, how many cm per minute will the depth of the water increase?

81

4Let’s learn the basics⑴

2

Special kinds of parallelograms, Parallel lines and area

Conditions for becoming special kinds of parallelograms

Conditions for a parallelogram to be a rectangle①

One angle is a right angle.

②

The lengths of the diagonals are equal.

⑵

rectangle
parallelogram

Conditions for a parallelogram to become a rhombus①

A pair of adjoining sides are equal in length.

②

The diagonals intersect perpendicularly.

⑶

parallelogram

rhombussquarerhombussquarerectangle

A parallelogram meeting the conditions for both a rectangle and a rhombus is called a square.

4 Prove that “A parallelogram with one□
right angle is a rectangle.”

ABCD on the left, assume that ∠A is 90°.

InWhen point E is on the extension of DA, since AB™DC,＝∠EAB＝90°.
Since both pairs of opposite angles in a parallelogram
are equal in size,∠A＝＝90° andTo sum up, ∠A＝＝＝＝90°.

＝＝90°,

which means four angles are equal; therefore,

ABCD

is a rectangle.

5 Prove that “A parallelogram whose□

diagonals are equal in length is aIn

ABCD on the left, assume that AC is equal to BDin length.

rectangle.”

In △ABC and △DCB,,

as shown in the hypothesis, AC＝since both pairs of opposite sides in a parallelogram＝DC,are equal in length,＝and since it is shared by both triangles,.

The statements above mean that all three pairs ofsides are equal in length, so△ABC≡△DCB; therefore, ∠B＝.

Since both pairs of opposite angles in a parallelogram
are equal in size,∠A＝∠C and ∠B＝∠D,
and since the sum of interior angles in a quadrilateral is＝＝90°, which means

four angles are equal; therefore,

ABCD is a rectangle.

360°,

＝＝

6 Prove that “A parallelogram having two adjoining sides that are equal in length is a rhombus.”□InABCD on the right, assume that AB is equal to BC in length.

Since both pairs of opposite sides in a parallelogram are equal in length,
AB＝CD and BC＝DA.
Therefore,
Since

＝＝＝.
are equal,

ABCD is a rhombus.

143

Chapter 7 Data Analysis

Data Analysis

Chapter 7●Key points of study●

1

Quartile points and Box and Whisker Plot☞P 180Quartile points…When the values of a data set are arranged in order⑴←smalllarge→of size (from smallest to largest), three points can be defined that
divide the data into four equal parts. The points are, from the smallest
to the largest: the first quartile, the second quartile (median), and thethird quartile.
⑵

minimum second quartile maximum（median）
valuevalue
first quartile
third quartile

Interquartile range…The difference between the third quartile and the first quartile.
▶（Interquartile range）＝（third quartile）−（first quartile）
Box and whisker plot…summarizes the distribution of the data using the⑶5 points: minimum value, first quartile, median, third quartile, andmaximum value.

▼1 Quartile points and Box and Whisker Plot Exercises 扌 P 184
Let’s learn the basics

Quartile points ⑴1When the data values are arranged from smallest to largest, the median of the left half of the data

Chapter 7(including the minimum value) is the first quartile, the median of the right half of the data (including themaximum value) is the third quartile, the median of the entire data set is the second quartile.
The data on the right shows the time it takes (in minutes) for each

26

36

27

50of 11 students to get to school. Find the minimum value, the maximum

34322333 （Unit：min.）Problem

48

2740

value, and the quartile points of this data.
Solution

The data can be arranged from smallest to largest as follows.

23

26

2727

32

33

34

3640

48

50

minimumvalue

first quartile

second quartile

third quartile

maximumvalue

23

27

33

40

50

Answer（Unit：min.）See the table above.

1 Find the minimum value, maximum value and quartile points of the following data.
□⑴

316□⑵

55

62

180

50

91068

65

1859

60

51157

67

50

91764

50

1156

69

15（Unit : hour）

64（Unit : g）

Chapter 7 Data Analysis

Comprehension test for Chapter 7
1 The following data shows the individual commuting times for

24

7111524514

a group of 15 students. Answer the following questions.

1118

61220

109 （Unit：min.）⑴

5Arrange the data in the frequency distribution chart on
the right.

Frequency （Number of students）
Class（min.）（people）
at least less than⑵

Draw the corresponding histogram in the diagram on
the right.

⑶

0〜 5
5〜10
10〜15
15〜20
20〜25

5Total15

interquartile
range

range

0

Fill in the following information in the table below:

0 5 10 15 20 25（min.）minimum value, maximum
value, quartile points, interquartile range, and total range.
minimumvalue⑷

first quartile second quartile third quartile maximumvalue

Draw a box and whisker plot in the diagram below.

01020（min.）Application of Chapter 71□

Box and whisker plots on the right summarize the scores achieved bybasketball teams A, B, and C in their last 15 games. If you could select only
one of these teams to play in the upcoming tournament, which team would youchoose? Answer using letters A, B, or C, and explain the reasons for yourchoice.

（Score）
100
80
60
40
200A

BC185