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).The ship is 41.81 feet from the lighthouse operator at the top of the lighthouse.

To find the distance from the lighthouse operator at the top of the lighthouse to the ship using the angle of elevation and the height of the lighthouse, we can use the sine ratio.

The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, we have:

• The angle of elevation (θ) = 73°
• The height of the lighthouse (opposite side) = 40 feet
• The distance from the lighthouse operator to the ship (hypotenuse) = d

The sine ratio is given by:

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

Substituting the known values:

\[ \sin(73°) = \frac{40}{d} \]

To find \( d \), we can rearrange the equation:

\[ d = \frac{40}{\sin(73°)} \]

Now, calculate \( \sin(73°) \) using a calculator:

\[ \sin(73°) \approx 0.9563 \]

Now we plug in the value:

\[ d \approx \frac{40}{0.9563} \approx 41.81 \text{ feet} \]

So, the distance from the lighthouse operator to the ship is approximately 41.81 feet (rounded to the nearest hundredth place).

In conclusion, the calculation shows that the distance found using the sine ratio is indeed the stated value of 41.81 feet.