Question

A telephone tower casts a 50-foot shadow that makes a 48° angle with the ground. If Fiona wants to find the length of the cable that secures the tower to the ground, how should she set up the problem?Option #1:  cos48°=x50 Option #2:  cos48°=50x Option #3:  cos50°=50x

Answer 1

To find the length of the cable that secures the tower to the ground, we need to understand the geometry involved in the situation. The telephone tower, the ground, and the cable form a right triangle, where:

- The height of the tower represents the opposite side of the angle.
- The length of the shadow represents the adjacent side of the angle (50 feet).
- The cable is the hypotenuse of the triangle.

Given that the angle formed by the ground and the tower’s shadow (48°) is involved, we can use the cosine function, which relates the adjacent side to the hypotenuse.

The cosine function is defined as:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]

In your options:

1. Option #1: \( \cos(48°) = \frac{x}{50} \) (where \( x \) is the length of the cable)
2. Option #2: \( \cos(48°) = \frac{50}{x} \)
3. Option #3: \( \cos(50°) = \frac{50}{x} \)

Since we know that the adjacent side is 50 feet and we want to find the length of the hypotenuse (the cable length \( x \)), we can set it up as follows:

\[
\cos(48°) = \frac{50}{x}
\]

This means we want to rearrange it to find \( x \):

Thus:

\[
x = \frac{50}{\cos(48°)}
\]

When looking for the correct answer among the provided options, **Option #2: \( \cos(48°) = \frac{50}{x} \)** best represents the situation, as it correctly relates the length of the shadow to the length of the cable.

So the answer is **Option #2**.