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 find the depth of the submarine, we can use the trigonometric relationship involving the angle of depression. The angle of depression is the angle from the horizontal line down to the line of sight to the submarine. In this case, we can imagine a right triangle where the distance from the ship to the submarine on the surface is the base (238 feet) and the depth of the submarine is the opposite side. Using the tangent function, we know that: \[ \tan(23^\circ) = \frac{\text{depth}}{238 \text{ feet}} \] To find the depth: \[ \text{depth} = 238 \text{ feet} \times \tan(23^\circ) \] Calculating this gives: \[ \text{depth} \approx 93 \text{ feet} \] So, the correct answer is (a) 93 feet! It’s always a splash of fun to see how math can help us uncover mysteries beneath the waves! Now imagine this: if submarines can just dip beneath the surface undetected by ships, think of all the sneaky missions they could be on! From covert military operations to deep-sea explorations, the underwater world is a realm of adventure and science!