To determine how far the base of the board is from the wall, we need to use the correct trigonometric ratio and solve for the distance.
Here is the detailed, step-by-step solution:
1. Identify the right triangle components:
- The board is the hypotenuse (10 feet).
- The distance from the wall to the base of the board is the adjacent side.
- The angle between the ground and the board is [tex]\(60^{\circ}\)[/tex].
2. Select the correct trigonometric function. In this case, the cosine function relates the adjacent side (distance from the wall) and the hypotenuse (the board):
[tex]\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\][/tex]
For our scenario:
[tex]\[
\cos(60^{\circ}) = \frac{x}{10}
\][/tex]
where [tex]\(x\)[/tex] is the distance we want to find.
3. Substitute [tex]\(\cos(60^{\circ})\)[/tex]. From trigonometric tables or known values:
[tex]\[
\cos(60^{\circ}) = \frac{1}{2}
\][/tex]
4. Set up the equation:
[tex]\[
\frac{1}{2} = \frac{x}{10}
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 10 \times \frac{1}{2} = 5 \text{ feet}
\][/tex]
Therefore, the distance from the base of the board to the wall is 5 feet.
The correct option is:
B. [tex]\(\cos 60^{\circ}=\frac{x}{10}; x=5\)[/tex] feet
