Refer to the following diagram for the roof of a house. In the figure, *x* is the length of a rafter measured from the top of a wall to the top of the roof; 𝜃 is the acute angle between a rafter and the horizontal; and *h* is the vertical distance from the top of the wall to the top of the roof.

Suppose that 𝜃 = 31° and *h* = 32.5 ft.

(a) Determine *x*. Round the answer to one decimal place.
*x* =  ft

(b) Find the area of the gable. Round the final answer to one decimal place. [The *gable* is the triangular region bounded by the rafters and the attic floor.]
ft2

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.

Refer to the following diagram for the roof of a house. In the figure, *x* is the length of a rafter measured from the top of a wall to the top of the roof; 𝜃 is the acute angle between a rafter and the horizontal; and *h* is the vertical distance from the top of the wall to the top of the roof.
Suppose that 𝜃 = 31° and *h* = 32.5 ft.
(a) Determine *x*. Round the answer to one decimal place.
(b) Find the area of the gable. Round the final answer to one decimal place.

Learn how to determine the length of a rafter and calculate the area of a gable using trigonometry. Given 𝜃 = 31° and h = 32.5 ft, we solve for the rafter length and gable area. Suitable for high school students (Grades 9-12).

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