We are given a diagram of the roof of a house with certain known values and need to find:

• (a) The length of a rafter (x) using the given angle (𝜃) and vertical height (h).
• (b) The area of the gable, which is the triangular region bounded by the rafters and the attic floor.

(a) Finding x

The triangle formed by the roof rafters is a right triangle, where:

• x is the hypotenuse (length of the rafter),
• h is the vertical leg (opposite the angle 𝜃),
• and the acute angle 𝜃 = 31° is between the rafter and the horizontal.

We can use the sine function because: $\sin(𝜃) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{x}$ Thus: $x = \frac{h}{\sin(𝜃)}$ Substitute the known values: $x = \frac{32.5 \, \text{ft}}{\sin(31^\circ)}$

Let’s calculate x.

(b) Finding the area of the gable

The gable forms a right triangle, and the area of a triangle is given by: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ Here, the base of the triangle is the horizontal leg (let's call it b), and the height is h.

To find b, we use the cosine function: $\cos(𝜃) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{x}$ Thus: $b = x \cos(𝜃)$ Once we have b, we can compute the area: $\text{Area} = \frac{1}{2} \times b \times h$

Let’s calculate b and the area.### Results:

(a) The length of the rafter (x) is approximately $63.1 \, \text{ft}$.

(b) The area of the gable is approximately $878.9 \, \text{ft}^2$.

Do you need more details, or have any other questions?

Here are some related questions you might consider:

1. How would the value of x change if the angle 𝜃 increased?
2. What is the relationship between the height h and the area of the gable?
3. Can you calculate the perimeter of the gable triangle?
4. What is the formula for the area of a triangle in general?
5. How would the area change if the height h were doubled?

Tip: When working with right triangles, always consider the sine, cosine, or tangent functions based on the sides and angle you have.