Question

A rectangle and a square have the same area. a. The width of the rectangle is w inches. The length of the rectangle is 15 inches less than 4 times the width of the rectangle. Write a polynomial to represent the area of the rectangle. b. The side of the square is w inches. Write a polynomial to represent the area of the square. c. Write an equation such that the area of the rectangle is equal to the area of the square. d. Solve the equation in part (c) to find the side length of the square. e. What are the length and the width of the rectangle?

Answer 1

9514 1404 393

Answer:

a) w(4w-15)

b) w²

c) w(4w -15) = w²

d) w = 5

e) 5 by 5

Explanation:

a) If w is the width, and the length is 15 less than 4 times the width, then the length is 4w-15. The area is the product of length and width.

A = w(4w -15)

__

b) If w is the side length, the area of the square is (also) the product of length and width:

A = w²

__

c) Equating the expressions for area, we have ...

w(4w -15) = w²

__

d) we can subtract the right side to get ...

4w² -15w -w² = 0

3w(w -5) = 0

This has solutions w=0 and w=5. Only the positive solution is sensible in this problem.

The side length of the square is 5 units.

__

e) The rectangle is 5 units wide, and 4(5)-15 = 5 units long.

The rectangle and square have the same width and the same area, so the rectangle must be a square.