Question
A wooden board is leaning against a house. The base of the board is 10 feet from the base of the house, and the base of the bo forms a \( 35^{\circ} \) angle with the ground. What is the length of the wooden board? Enter your answer, rounded to the nearest tenth, in the box
Ask by Fitzgerald Christensen. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The length of the wooden board is approximately 12.2 feet.
Solution
To determine the length of the wooden board, we can use trigonometry, specifically the cosine function.
Given:
- The distance from the base of the board to the house (adjacent side) is 10 feet.
- The angle between the ground and the board is \( 35^{\circ} \).
The cosine of the angle is defined as the adjacent side divided by the hypotenuse (the length of the board, which we'll call \( L \)):
\[
\cos(35^{\circ}) = \frac{10}{L}
\]
Solving for \( L \):
\[
L = \frac{10}{\cos(35^{\circ})}
\]
Calculating \( \cos(35^{\circ}) \):
\[
\cos(35^{\circ}) \approx 0.8192
\]
Now, substitute back to find \( L \):
\[
L = \frac{10}{0.8192} \approx 12.2168 \text{ feet}
\]
Rounding to the nearest tenth:
\[
L \approx 12.2 \text{ feet}
\]
**Answer:**
12.2
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Simplify this solution The Deep Dive
To find the length of the wooden board, we can use trigonometry. Since the board forms a \( 35^{\circ} \) angle with the ground, we can use the cosine function. Cosine of the angle is equal to the adjacent side (distance from the house) divided by the hypotenuse (length of the board): \[ \cos(35^{\circ}) = \frac{10}{\text{Length of the board}} \] Rearranging this gives us: \[ \text{Length of the board} = \frac{10}{\cos(35^{\circ})} \] Using a calculator, we find that \( \cos(35^{\circ}) \approx 0.8192\). Thus: \[ \text{Length of the board} \approx \frac{10}{0.8192} \approx 12.2 \text{ feet} \] So, the length of the wooden board is approximately **12.2 feet**.
