A ship is on the surface of the water, and its radar detects a submarine 238 feet away, at an angle of depression of \( 23 \circ \). How deep underwater is the submarine? a) 93 feet b) 219 feet c) 42 feet d) 101 feet e) 209 feet

To determine the depth of the submarine, we can use trigonometry, specifically the sine function in a right triangle. **Given:** - **Distance (hypotenuse)** from the ship to the submarine: 238 feet - **Angle of depression**: \( 23^\circ \) **Objective:** Find the depth (\( d \)) of the submarine underwater. **Approach:** 1. **Understand the Scenario:** - The angle of depression is measured from the horizontal line of sight downward to the submarine. - This forms a right triangle where: - The **hypotenuse** is the line of sight (238 feet). - The **opposite side** is the depth of the submarine (\( d \)). - The **adjacent side** is the horizontal distance from the ship to the point directly above the submarine. 2. **Use the Sine Function:** \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] \[ \sin(23^\circ) = \frac{d}{238} \] 3. **Solve for Depth (\( d \)):** \[ d = 238 \times \sin(23^\circ) \] 4. **Calculate \( \sin(23^\circ) \):** \[ \sin(23^\circ) \approx 0.3907 \] 5. **Compute the Depth:** \[ d = 238 \times 0.3907 \approx 93 \text{ feet} \] **Answer:** **a) 93 feet**