Let's break down the problem step by step to determine which equation Kim can use to find [tex]\( x \)[/tex], the length of the pole.
1. Understand the context:
- The pole casts a 12-foot shadow.
- The angle of elevation of the sun is [tex]\( 40^\circ \)[/tex].
2. Set up the problem:
- The pole, its shadow, and the angle of elevation create a right-angled triangle.
- In this triangle:
- The height of the pole [tex]\( x \)[/tex] is the side opposite the angle of elevation.
- The shadow is the side adjacent to the angle of elevation.
- The angle of elevation is [tex]\( 40^\circ \)[/tex].
3. Use the appropriate trigonometric function:
- The tangent function relates the opposite side and the adjacent side in a right-angled triangle:
[tex]\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- For our given angle of elevation [tex]\( 40^\circ \)[/tex]:
[tex]\[
\tan(40^\circ) = \frac{x}{12}
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- Multiply both sides by 12 to isolate [tex]\( x \)[/tex]:
[tex]\[
x = 12 \cdot \tan(40^\circ)
\][/tex]
5. Evaluate the expression to find [tex]\( x \)[/tex]:
- After calculating [tex]\( 12 \cdot \tan(40^\circ) \)[/tex], we get a numerical result of approximately 10.0692.
6. Identify the corresponding equation:
- Among the given choices:
- [tex]\(\frac{\sin 40}{x}=\frac{\sin 60}{12}\)[/tex]
- [tex]\(\frac{\sin 40}{12}=\frac{\sin 60}{x}\)[/tex]
- [tex]\(\frac{\sin 60}{x}=\frac{\sin 80}{12}\)[/tex]
- None of these choices directly match our initial equation using tangent. However, considering the relationship setup and the solving process, the correct equation from the choice is:
- [tex]\[
\frac{\tan(40^\circ)}{1} = \frac{x}{12}
\][/tex]
- Then rearrange it to:
- [tex]\[
x = \tan(40^\circ) \cdot 12
\][/tex]
After following these steps, the appropriate choice for Kim to use to find the height of the pole, [tex]\( x \)[/tex], would be the equation incorporating tangent:
[tex]\[
\text{tan}(40^\circ) \cdot 12 = x
\][/tex]
Therefore, this corresponds to the correct interpretation which led us to choosing the second numeric confirmation. This calculation yields the correct height, [tex]\( x \)[/tex], of approximately 10.0692 feet.
