A contractor is building a porch along two sides of a house. The house is rectangular with a width of 32...

Question

14. A contractor is building a porch along two sides of a house. The house is rectangular with a width of 32 feet and a length of 50 feet. The porch will have the same width on each side of the house.
a. Write a polynomial that represents the combined area of the first floor of the house and the porch.
b. The owners want the combined area of the first floor and the porch to be 2320 square feet. How wide should the contractor build the porch?

SnapAndSolve AI · Accepted Answer

a. Polynomial for the Combined Area

Let x represent the width of the porch, in feet. Since the porch runs along two adjacent sides of the house, the combined rectangular dimensions are (32 + x) feet by (50 + x) feet.

The combined area of the house’s first floor and the porch is:

A(x) = (32 + x)(50 + x)

Expanding:

A(x) = 1600 + 32x + 50x + x²

A(x) = x² + 82x + 1600 square feet

b. Width of the Porch

Set the combined area equal to 2320 square feet:

x² + 82x + 1600 = 2320

Subtract 2320 from both sides:

x² + 82x − 720 = 0

Factor the polynomial:

(x + 90)(x − 8) = 0

Therefore, x = −90 or x = 8. A width cannot be negative, so discard −90.

The contractor should build the porch 8 feet wide.

Check: (32 + 8)(50 + 8) = 40 × 58 = 2320 square feet.

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A contractor is building a porch along two sides of a house. The house is rectangular with a width of 32...

Answer

a. Polynomial for the Combined Area

b. Width of the Porch

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