To solve this problem, we need to find the distance from the base of the board to the wall using trigonometric ratios. We are given the length of the board and the angle it makes with the ground.
Here are the steps to find the correct ratio and the distance:
1. Identify the given values:
- Length of the board [tex]\( L = 10 \)[/tex] feet.
- Angle with the ground [tex]\( \theta = 60^\circ \)[/tex].
2. Determine the appropriate trigonometric ratio:
To find the distance from the base of the board to the wall (denoted as [tex]\( x \)[/tex]), we notice that this distance corresponds to the adjacent side of the right triangle formed by the board, the wall, and the ground. Therefore, the cosine function is appropriate:
[tex]\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{10}
\][/tex]
3. Use the given angle [tex]\( \theta = 60^\circ \)[/tex]:
[tex]\[
\cos(60^\circ) = \frac{x}{10}
\][/tex]
4. Calculate [tex]\( \cos(60^\circ) \)[/tex]:
[tex]\[
\cos(60^\circ) = \frac{1}{2}
\][/tex]
5. Set up the equation:
[tex]\[
\frac{1}{2} = \frac{x}{10}
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 10 \times \frac{1}{2} = 5 \text{ feet}
\][/tex]
Thus, the distance from the base of the board to the wall is [tex]\( 5 \)[/tex] feet.
Now, let's review the given choices:
- A. [tex]\( \sin 60^\circ = \frac{x}{10} ; x \approx 8.66 \text{ feet} \)[/tex]
- This option uses the sine function incorrectly. Incorrect.
- B. [tex]\( \cos 60^\circ = \frac{10}{x} ; x = 20 \text{ feet} \)[/tex]
- This option misappropriates the cosine function. Incorrect.
- C. [tex]\( \cos 60^\circ = \frac{x}{10} ; x = 5 \text{ feet} \)[/tex]
- This option correctly uses the cosine function with the right calculation. Correct.
- D. [tex]\( \sin 60^\circ = \frac{10}{x} ; x \approx 11.5 \text{ feet} \)[/tex]
- This option incorrectly applies the sine function. Incorrect.
The correct answer is:
C. [tex]\( \cos 60^\circ = \frac{x}{10} ; x = 5 \text{ feet} \)[/tex]
