A homeowner wants to build a circular fire pit an outdoor patio. The pit will be surrounded by a brick ledge. The diameter of the pit must be at least 16 inches. The ledge myust have a width of at least 2 inches which is the width of each brick. The maximum area for the pit and the ledge is 2826 square inches. What are the possible values for w, the width of the ledge the homeowner could build?

The possible values of the width of ledge w are 2 in, 4 in, 6 in, 8 in, 10 in, 12 in, 14 in, 16 in, 18 in, 20 in, 22 in.

Step-by-step explanation:

Since the area of he pit and width of ledge are in a circular pattern, the area of the pit + ledge is given by

A = π(R² - r²) where R = distance from center of pit to outer part of ledge and r = radius of pit.

Since diameter of pit = 16 in, radius of pit, r = 16 in/2 = 8 in.

Also, the maximum area of the pit and ledge is 2826 in². So, A = 2826 in²

Since A = π(R² - r²)

R = √(A/π + r²)

substituting th values of A and r into R, we have

R = √(2826 in²/π + (8 in)²)

R = √(899.54 in² + 64 in²)

R = √963.54 in²

R = 31.04 in

R ≅ 31 in

Now, the maximum width of the ledge w = R - r = 31 - 8 = 23 in.

Since the width of each brick is 2 in, the width of the ledge must be in multiples of 2. Since the maximum width of the ledge is 23 in, we divide it by 2 in to know how many bricks are required for the width of the ledge.

So, number of bricks = 23 in/2 in = 11.5 ≅ 12

So, we require from 1 to 12 bricks to have a minimum to maximum width of ledge.

Since the width of ledge = no of bricks × width of brick

So, the possible values of w are 1 × 2 in, 2 × 2 in, 3 × 2 in, 4 × 2 in, 5 × 2 in, 6 × 2 in, 7 × 2 in, 8 × 2 in, 9 × 2 in, 10 × 2 in, 11 × 2 in

So, the possible values of the width of ledge w are 2 in, 4 in, 6 in, 8 in, 10 in, 12 in, 14 in, 16 in, 18 in, 20 in, 22 in.

We do not include 24 in since the maximum width of the ledge is 23 in.

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