To solve the problem using the sine ratio, we can set up the relationship involving the height of the lighthouse and the angle of depression.
Let's denote:
- \( h = 40 \) feet (the height of the lighthouse)
- \( \theta = 17^\circ \) (the angle of depression)
In a right triangle formed by the height of the lighthouse, the distance from the base of the lighthouse to the ship, and the line of sight from the top of the lighthouse to the ship, we can apply the sine function.
The sine of the angle of depression can be defined as:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
In this case, the "opposite" side corresponds to the height of the lighthouse (40 feet), and the "hypotenuse" will be the distance from the top of the lighthouse to the ship. Rearranging the sine function gives:
\[ \text{hypotenuse} = \frac{\text{opposite}}{\sin(\theta)} = \frac{40}{\sin(17^\circ)} \]
However, we need to find how far away the ship is (the horizontal distance, let's call it \( d \)). To do this, we will use:
\[ \sin(17^\circ) = \frac{h}{d} \]
or
\[ d = \frac{h}{\sin(17^\circ)} \]
Now let's calculate it. First, we find \(\sin(17^\circ)\):
Using a calculator for \(\sin(17^\circ)\):
\[ \sin(17^\circ) \approx 0.2924 \]
Now substituting it into the formula for \( d \):
\[ d = \frac{40}{\sin(17^\circ)} = \frac{40}{0.2924} \approx 136.8 \]
Thus, the distance from the base of the lighthouse to the ship is approximately 136.8 feet.
Therefore, the correct response is 136.8 feet.