A submarine, cruising at a depth $d = 65$ meters, is on a trajectory that passes directly below a ship (see figure). If $\theta$ is the angle of depression from the ship to the submarine, find the distance $L$ from the ship to the sub when $\theta = 70 ^ { \circ }$. Round to the nearest meter.

A) L = 0 meters
B) L = 84 meters
C) L = 69 meters
D) L = 190 meters
E) L = 24 meters

Question

A submarine, cruising at a depth $d = 65$ meters, is on a trajectory that passes directly below a ship (see figure). If $\theta$ is the angle of depression from the ship to the submarine, find the distance $L$ from the ship to the sub when $\theta = 70 ^ { \circ }$. Round to the nearest meter.

A) L = 0 meters 
B) L = 84 meters 
C) L = 69 meters 
D) L = 190 meters 
E) L = 24 meters

Quizplus · Accepted Answer

In the given figure, we can see that the distance from the ship to the point directly above the submarine is $d = 65m$.

Now, we can use the trigonometric relationship between the angle of depression and the horizontal distance to find $L$.

Using the trigonometric function $\tan$, we have:

$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{L}{d}$

Substituting $d = 65m$ and $\theta = 70^{\circ}$, we get:

$\tan(70^{\circ}) = \frac{L}{65}$

Solving for $L$, we get:

$L = 65 \tan(70^{\circ}) \approx 69m$

Therefore, the distance $L$ from the ship to the sub when $\theta = 70^{\circ}$ is approximately 69 meters, which is answer choice C.

A Submarine, Cruising at a Depth d=65d = 65d=65 Meters, Is on a Trajectory That Passes Directly Below a Is

A Submarine, Cruising at a Depth $d = 65$ Meters, Is on a Trajectory That Passes Directly Below a Is