Question

A lighthouse operator is sending signals to a ship with a 17°
17
°
angle of depression. The height of the lighthouse is 40 feet relative to sea level. Which of the following correctly applies the sine ratio to find how far away the ship is from the top of the lighthouse? Round to the nearest tenth.(1 point)
Responses

96.8 feet
96.8 feet

157.5 feet
157.5 feet

130.8 feet
130.8 feet

136.8 feet

Answer 1

To find how far away the ship is from the base of the lighthouse (the horizontal distance), we can use the sine ratio from trigonometry.

1. **Understand the scenario**:
- Height of the lighthouse (opposite side in the triangle) = 40 feet
- Angle of depression from the top of the lighthouse to the ship = 17°
- Therefore, the angle of elevation from the ship to the top of the lighthouse is also 17°.

2. **Set up the relationship**:
- We use the sine function, which is defined as:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

3. **Apply it to our scenario**:
However, we want to find the adjacent side (distance from the base of the lighthouse to the ship), so we will rearrange the sine function to solve for the hypotenuse first:

\[
\sin(17°) = \frac{40}{\text{hypotenuse}}
\]

Rearranging gives us:

\[
\text{hypotenuse} = \frac{40}{\sin(17°)}
\]

4. **Calculate the hypotenuse**:
- First, calculate \(\sin(17°)\):

Using a calculator:

\[
\sin(17°) \approx 0.2924
\]

Now substitute back:

\[
\text{hypotenuse} = \frac{40}{0.2924} \approx 136.8 \text{ feet}
\]

Thus, the distance from the top of the lighthouse to the ship (hypotenuse) is approximately **136.8 feet**.

The correct answer from your given options is:

**136.8 feet**.