From the top of a 224-ft lighthouse, the angle of depression to a ship in the ocean is 23°. How far is the ship from the base of the lighthouse? Give your answer to the nearest foot.

To find the distance from the ship to the base of the lighthouse, we can use trigonometry. The problem involves a right triangle where:

- The height of the lighthouse is the opposite side (\( O \)) of the triangle.
- The distance from the ship to the base of the lighthouse is the adjacent side (\( A \)).
- The angle of depression is 23°.

We can use the tangent function, which relates the opposite side to the adjacent side in a right triangle:

\[
\tan(\theta) = \frac{O}{A}
\]

Where:
- \( \theta = 23^\circ \) (angle of depression),
- \( O = 224 \) feet (height of the lighthouse).

Rearranging the formula to solve for \( A \) (the distance from the ship to the base of the lighthouse):

\[
A = \frac{O}{\tan(\theta)}
\]

Substitute the known values:

\[
A = \frac{224}{\tan(23^\circ)}
\]

Now, let's calculate:

\[
A \approx \frac{224}{\tan(23^\circ)} \approx \frac{224}{0.4245} \approx 527.63 \text{ feet}
\]

Rounding to the nearest foot:

\[
A \approx 528 \text{ feet}
\]

So, the ship is approximately **528 feet** from the base of the lighthouse.

Would you like further details or have any questions?

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**Relative Questions:**
1. How would the distance change if the angle of depression increased to 30°?
2. What if the lighthouse was 300 feet tall instead of 224 feet?
3. How does the angle of depression relate to the height of the lighthouse in such problems?
4. What trigonometric function would you use if the angle of elevation was given instead?
5. How would the solution change if the height of the lighthouse were measured in meters?

**Tip:** Always double-check your angle mode (degrees vs. radians) when performing trigonometric calculations on a calculator.

Explore how to calculate the distance from a ship to the base of a 224-ft lighthouse using trigonometry. Detailed step-by-step solution involving the tangent function and right triangle geometry. Answer provided to the nearest foot.

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