Question

as the diagram shows below, submarines use sonar systems, which are similar to radar systems, to detect obstacles. sonar systems use sound to detect objects under water.
you are traveling underwater in a submarine. the sonar system detects an iceberg 4000 meters ahead, with an angle of depression of 34° to the bottom of the iceberg. how many meters must the submarine lower to pass under the iceberg? round to the nearest meter.
not drawn to scale
4000 m
34
19°
1500 m

Explanation:

Step1: Find the depth from the first submarine to the bottom of the iceberg

We use the tangent function. The tangent of an angle in a right - triangle is given by $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.
Let $h_1$ be the depth from the first submarine to the bottom of the iceberg. The adjacent side is $4000$ m and the angle of depression is $34^{\circ}$. Since the angle of depression is equal to the angle of elevation in the right - triangle formed, we have $\tan34^{\circ}=\frac{h_1}{4000}$.
So, $h_1 = 4000\times\tan34^{\circ}$.
Using a calculator, $\tan34^{\circ}\approx0.6745$, then $h_1=4000\times0.6745 = 2698$ m.

Step2: Find the depth from the second submarine to the bottom of the iceberg

Let $h_2$ be the depth from the second submarine to the bottom of the iceberg. The adjacent side is $1500$ m and the angle is $19^{\circ}$.
Using the tangent function $\tan19^{\circ}=\frac{h_2}{1500}$.
So, $h_2 = 1500\times\tan19^{\circ}$.
Using a calculator, $\tan19^{\circ}\approx0.3443$, then $h_2=1500\times0.3443=516.45$ m.

Step3: Calculate the difference in depth

The distance the submarine must lower is $h_1 - h_2$.
$h_1 - h_2=2698 - 516.45=2181.55\approx2182$ m.

Answer:

$2182$ m