Question

select the correct answer.

a model of stevens house is shown in the diagram. he is replacing the roof, which consists of two rectangular surfaces. the diagram shows one side of the roof, with the other side hidden from view. what is the total area of the roof to the nearest square foot?

1,141 square feet
1,397 square feet
1,976 square feet
2,794 square feet

Explanation:

Identify the roof dimensions

The roof consists of two identical rectangular surfaces.
The length of each rectangular surface is given as \(38\text{ ft}\).
The width of the house is \(26\text{ ft}\).
The roof forms a symmetric gable, and the angle of elevation of the roof is \(45^\circ\).
This forms an isosceles right triangle at the top gable if we consider the peak, or two symmetric \(45^\circ\)-\(45^\circ\)-\(90^\circ\) right triangles.

Calculate the roof slant length

Using the Isosceles Right Triangle and Special Right Triangles knowledge points

$$
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Calculate the area of one rectangular roof surface

The area of one side of the roof is the product of its length and its slant height.

$$ \text{Area}_{\text{one side}} = 38 \times 13\sqrt{2} \approx 38 \times 18.385 = 698.63\text{ sq ft} $$

Calculate the total roof area

The total roof consists of two identical rectangular surfaces.

$$ \text{Total Area} = 2 \times (38 \times 13\sqrt{2}) = 988\sqrt{2} \approx 1397.24\text{ sq ft} $$

Round to the nearest square foot

Rounding \(1397.24\) to the nearest whole number gives \(1,397\text{ square feet}\).

Answer:

• 1,141 square feet
• 1,397 square feet (Correct answer)
• 1,976 square feet
• 2,794 square feet