A square garden plot has a side length of 3s - 2 yards. A walkway of uniform width s + 3 y… — ACT

A square garden plot has a side length of $3s - 2$ yards. A walkway of uniform width $s + 3$ yards is built around the entire garden. Which of the following expressions represents the area, in square yards, of the walkway?

A
7s^2 + 20s - 3
B
16s^2 + 28s + 20
16s^2 + 52s + 12Answer
D
16s^2 + 12
E
4s^2 + 52s + 12

Answer

16s^2 + 52s + 12

The correct answer is found by first calculating the outer side length, which is the inner side length plus twice the walkway width:

(3s - 2) + 2(s + 3) = 5s + 4

. Squaring this yields the outer area of

25s^2 + 40s + 16

. The inner area is

(3s - 2)^2 = 9s^2 - 12s + 4

. Subtracting the inner area from the outer area and distributing the negative sign properly gives

(25s^2 + 40s + 16) - (9s^2 - 12s + 4) = 16s^2 + 52s + 12

Step-by-Step Solution

Determine the outer side length of the square including the walkway.

The outer side length is

(3s - 2) + 2(s + 3) = 3s - 2 + 2s + 6 = 5s + 4

yards.

Since the walkway surrounds the garden on all sides, the width of the walkway must be added to both ends of the garden's side length.

Calculate the area of the outer square and the inner square garden by squaring their respective side lengths.

Outer Area =

(5s + 4)^2 = 25s^2 + 40s + 16

and Inner Area =

(3s - 2)^2 = 9s^2 - 12s + 4

The area of a square is equal to the square of its side length.

Subtract the inner garden area from the outer area to find the walkway area, distributing the negative sign to all terms of the inner area.

Walkway Area =

(25s^2 + 40s + 16) - (9s^2 - 12s + 4) = 25s^2 + 40s + 16 - 9s^2 + 12s - 4 = 16s^2 + 52s + 12

square yards.

The area of the walkway is the difference between the total outer area and the inner garden area.

Key Concept

Operations on Polynomials

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