Question

write and simplify an expression for each situation.
you are planning to put a pool in your backyard. the length of the pool is twice the width. you also will have a 2 foot wide deck around the pool

a. write an expression for the area of the pool.

b. write an expression for the area of the deck and pool.

c. write and simplify an expression for the area of the deck.

Explanation:

Part (a)

Step1: Define variables

Let the width of the pool be \( w \). Then the length of the pool, which is twice the width, is \( l = 2w \).

Step2: Use area formula for rectangle

The area of a rectangle (pool) is \( A=\text{length}\times\text{width} \). Substituting \( l = 2w \) and width \( w \), we get \( A = 2w\times w=2w^{2} \).

Step1: Determine dimensions with deck

The deck is 2 feet wide around the pool. So the new width (pool + deck) is \( w + 2+ 2=w + 4 \) (2 feet on each side). The new length (pool + deck) is \( 2w+ 2 + 2=2w + 4 \) (2 feet on each side).

Step2: Calculate area of deck and pool

Using the area formula for a rectangle, \( A=\text{length}\times\text{width} \). Substituting the new length \( 2w + 4 \) and new width \( w + 4 \), we get \( A=(2w + 4)(w + 4) \).

Step1: Recall areas

We know the area of the pool is \( 2w^{2} \) (from part a) and the area of the deck and pool is \( (2w + 4)(w + 4) \) (from part b).

Step2: Expand the area of deck and pool

First, expand \( (2w + 4)(w + 4) \):

$$
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$$

Step3: Subtract pool area from total area

The area of the deck is the area of (deck + pool) minus the area of the pool. So \( A=(2w^{2}+12w + 16)-2w^{2} \).

Step4: Simplify the expression

Simplifying \( (2w^{2}+12w + 16)-2w^{2} \), the \( 2w^{2} \) terms cancel out, giving \( 12w + 16 \).

Answer:

The expression for the area of the pool is \( 2w^{2} \) (where \( w \) is the width of the pool in feet).

Part (b)