Question

A lighthouse operator is sending signals to a ship. The angle of elevation from the ship to the lighthouse operator is 73°. The height of the lighthouse is 40 feet relative to sea level. Which of the following correctly applies the sine ratio to find the distance from the lighthouse operator on top of the light house to the ship? Round to the nearest hundredth place (2 decimal places). x feet 40 feet 73° (1 point) The ship is feet from the lighthouse operator at the top of the lighthouse.

Answer 1

To solve for the distance from the lighthouse operator to the ship, we will use the sine ratio from trigonometry. The sine ratio relates the opposite side of a right triangle to the hypotenuse. In this scenario:

- The height of the lighthouse is the opposite side (40 feet).
- The distance from the lighthouse operator to the ship is the hypotenuse (denote this as \( x \)).
- The angle of elevation (the angle from the ship to the top of the lighthouse) is 73°.

The sine function is defined as:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

For our situation, we can write the equation as follows:
\[
\sin(73°) = \frac{40}{x}
\]

To solve for \( x \), we can rearrange this equation:
\[
x = \frac{40}{\sin(73°)}
\]

Next, we will calculate \( \sin(73°) \). Typically, you would use a calculator to find this value.

Using a calculator:
\[
\sin(73°) \approx 0.9563
\]

Now we can substitute this value back into our equation to find \( x \):
\[
x = \frac{40}{0.9563} \approx 41.85
\]

Therefore, the distance from the lighthouse operator to the ship is approximately **41.85 feet**.

Final answer: The ship is approximately **41.85 feet** from the lighthouse operator at the top of the lighthouse.