A submarine traveling at 9.0 mph is descending at an angle of depression of 5". How many minutes, to the nearest tenth, does it take the submarine to reach a depth of 80 feet?

To solve this problem, we'll use trigonometry to determine how long it takes the submarine to reach a depth of 80 feet given its speed and angle of depression. Here's the step-by-step process:

### Step 1: Convert the angle to radians
The angle of depression is given as \( 5^\circ \). However, it's not necessary to convert it to radians for this problem since trigonometric functions can work directly with degrees.

### Step 2: Set up the trigonometric relationship
Since the angle of depression is the same as the angle formed by the path of the submarine and the horizontal, we can use the sine function to relate the depth (opposite side) and the hypotenuse (the distance traveled by the submarine).

\[
\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\text{depth}}{\text{distance traveled}}
\]
\[
\sin 5^\circ = \frac{80 \text{ feet}}{d}
\]

### Step 3: Solve for the distance traveled (\( d \))
\[
d = \frac{80 \text{ feet}}{\sin 5^\circ}
\]

Using a calculator to find \(\sin 5^\circ\):

\[
\sin 5^\circ \approx 0.0872
\]
\[
d \approx \frac{80 \text{ feet}}{0.0872} \approx 917.43 \text{ feet}
\]

### Step 4: Convert the distance to miles
Since the submarine's speed is given in miles per hour (mph), we should convert the distance from feet to miles. 

\[
\text{1 mile} = 5280 \text{ feet}
\]
\[
\text{Distance in miles} \approx \frac{917.43 \text{ feet}}{5280 \

Learn how to calculate the depth of a submarine descending at an angle of depression of 5 degrees. This problem involves trigonometric relationships and speed-distance calculations. Suitable for high school students.

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