Alright, let's go through the problem step by step.
1. Understanding the problem:
- We are given a pole that casts a 12-foot shadow.
- The sun's angle of elevation is \(40^\circ\).
- We need to find the equation that can help us determine \(x\), which is the length of the pole.
2. Identifying the right trigonometric function:
- In this problem, we are dealing with angles and the lengths of sides in a right triangle formed by the pole, its shadow, and the line of sight from the top of the pole to the tip of the shadow.
3. Using the Sine Law:
- The Sine Law states:
[tex]\[
\frac{\sin A}{a} = \frac{\sin B}{b}
\][/tex]
for any triangle, where \(A\) and \(B\) are angles, and \(a\) and \(b\) are the sides opposite those angles respectively.
4. Applying the given values:
- In the given problem, we need an equation to relate the angle of elevation, the shadow length, and the pole's height.
5. Choosing the correct equation:
- We need to identify which of the given equations appropriately uses the Sine Law to solve for \(x\) (the length of the pole):
- \(\frac{\sin 40}{x} = \frac{\sin 60}{12}\)
- \(\frac{\sin 40}{12} = \frac{\sin 60}{x}\)
- \(\frac{\sin 60}{x} = \frac{\sin 80}{12}\)
- \(\frac{\sin 80}{x} = \frac{\sin 40}{12}\)
6. Comparing the options:
- We are working with \(\sin 40^\circ\), \(\sin 60^\circ\), and the known length of the shadow \(12\), to find the height \(x\).
- Match given values with appropriate form:
[tex]\[
\frac{\sin 40}{12} = \frac{\sin 60}{x}
\][/tex]
This equation appropriately uses the Sine Law to relate the angle of elevation \(40^\circ\), the known length of shadow (12 feet), and another angle of \(60^\circ\) with the unknown length \(x\).
So, the correct equation that Kim could use to find \(x\), the length of the pole, is:
[tex]\[
\boxed{\frac{\sin 40}{12}=\frac{\sin 60}{x}}
\][/tex]
