John is building a rectangular fence around a garden. The length will be 5 feet shorter than its width. If the total area of the garden is 176 square feet, what is the length of the diagonal (in feet)?  Answers:
A) between 18 and 19
B) between 19 and 20
C) between 20 and 21
D) between 21 and 22
E) between 22 and 23

Rationale: to solve this , we first need to set up an equation for the area of the garden . if x is the width , then we have x ( x - 5 ) = 176 . by putting the equation in standard form , we get x ^ 2 - 5 x - 176 = 0 . by using the quadratic formula , we get roots of - 11 and 16 . we know that x is the width , and x - 5 is the length , so by using the roots , we get 16 as the width ( x ) , and 11 as the length ( x - 5 ) . once we have this , we can use the pythagorean theorem to solve for the diagonal . plugging in the length and width , we will get d ^ 2 = 16 ^ 2 + 11 ^ 2 = 256 + 121 = 377 the square root of 377 is less than 20 , since 20 ^ 2 = 400 . going down a number , we can compute 19 ^ 2 = 361 < 377 . therefore , the length of the diagonal must be between 19 and 20 . the correct answer is b .
Answer: B