A 20-ft by 40-ft swimming pool is surrounded by a deck d feet wide. What is the area of the deck?

area of pool = (20)(4) = 800 ft^2

length of pool plus deck = 40+2d
width of pool plus deck = 20+2d
Area of whole thing = (40+2d)(20+2d)
= 800 + 120d + 4d^2

area of deck = 4d^2 +120d + 800- 800
= 4d^2 + 120d

To find the area of the deck surrounding the swimming pool, we need to subtract the area of the pool from the area of the pool with the deck.

First, let's calculate the area of the pool. The pool has dimensions 20 ft by 40 ft, so its area is given by multiplying the length and width:

Area of pool = Length × Width = 20 ft × 40 ft = 800 square feet.

Now, let's find the total dimensions of the pool with the deck. Since the deck surrounds the pool on all sides, we need to add twice the width of the deck to the dimensions of the pool.

The length of the pool with the deck will be: 20 ft + 2d (adding the deck width on both sides)
The width of the pool with the deck will be: 40 ft + 2d (adding the deck width on both sides)

The area of the pool with the deck is then given by multiplying the length and width:

Area of pool with deck = (20 ft + 2d) × (40 ft + 2d).

Now, we can find the area of the deck by subtracting the area of the pool from the area of the pool with the deck:

Area of deck = Area of pool with deck - Area of pool.

Substituting the expressions we have:

Area of deck = (20 ft + 2d) × (40 ft + 2d) - 800 square feet.

The resulting expression is the area of the deck in terms of the deck width "d".