To solve this problem, we need to use the appropriate trigonometric ratio based on the given angle and the length of the board.
Given:
- The length of the board \( AB \) is 10 feet.
- The angle \( \theta \) between the board and the ground is \( 60^\circ \).
We need to find the horizontal distance \( x \) from the base of the board to the wall (point \( A \) to the wall at point \( C \)). This scenario forms a right triangle \( ABC \) where:
- \( AC \) (the distance we need to find) is the adjacent side of the angle \( 60^\circ \).
- \( AB \) (the length of the board) is the hypotenuse.
The cosine function relates the adjacent side to the hypotenuse in a right triangle. Specifically,
[tex]\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \][/tex]
Using the given values:
[tex]\[ \cos(60^\circ) = \frac{x}{10} \][/tex]
We know that \( \cos(60^\circ) \) is \( \frac{1}{2} \). Hence,
[tex]\[ \frac{1}{2} = \frac{x}{10} \][/tex]
To solve for \( x \), we multiply both sides by 10:
[tex]\[ x = 10 \times \frac{1}{2} \][/tex]
[tex]\[ x = 5 \][/tex]
Therefore, the base of the board (point \( A \)) is 5 feet away from the wall (point \( C \)).
The correct trig ratio and distance from the wall, according to the choices given, is:
C. [tex]\(\cos 60^\circ = \frac{x}{10} ; x=5\)[/tex] feet
