To find the length of the boards on the sides of the dormer, we can model the dormer as an isosceles triangle. The base of the triangle is 9.3 feet, and each of the two base angles is 70°. We want to calculate the length of the sides of the triangle (the sides of the dormer), which are the boards Everett needs to replace.

Here’s the step-by-step method:

Step 1: Identify the height of the triangle

Since the dormer forms an isosceles triangle, we can split it into two right triangles by drawing a height from the top vertex to the midpoint of the base. The base is 9.3 feet, so half of the base is:

$\text{Half of the base} = \frac{9.3}{2} = 4.65 \, \text{feet}$

The height, $h$, forms one leg of the right triangle, the half-base forms the other leg, and the side of the dormer forms the hypotenuse. The base angle of 70° is given, so we can use trigonometric ratios to find the length of the side (the hypotenuse).

Step 2: Use the cosine function

In the right triangle, the cosine of the angle relates the adjacent side (half of the base) to the hypotenuse (the side of the dormer). The cosine function is:

$\cos(70^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4.65}{L}$

Where $L$ is the length of the side of the dormer (the hypotenuse). Solving for $L$:

$L = \frac{4.65}{\cos(70^\circ)}$

Step 3: Calculate the value

We now calculate the value of $L$ using $\cos(70^\circ)$.

$L = \frac{4.65}{\cos(70^\circ)} = \frac{4.65}{0.3420} \approx 13.59 \, \text{feet}$

Final Answer:

The length of the boards on the sides of the dormer is approximately 13.59 feet.

Would you like further details or have any questions?

Relative Questions:

1. How can we calculate the height of the dormer using the given information?
2. How does the law of cosines apply in this scenario, and how could it be used instead?
3. If the base angle were different, how would it affect the length of the sides?
4. How can trigonometric functions help in real-life construction problems?
5. What if the dormer base was unknown—could we still calculate the side lengths with a different method?

Tip:

In problems involving triangles, especially isosceles or right triangles, trigonometric functions like sine, cosine, and tangent are powerful tools for solving side lengths or angles.