Question

Demetry is deep-sea fishing. They have a device that tells them the diagonal distance from their boat to the fish, and how far down the fish is under the water. If the device tells them there is a fish 109 feet down and 230 feet away from them, what is the approximate angle they’ll need to get their fishing line to catch the fish?

Answer 1

To find the angle at which Demetry needs to drop their fishing line to catch the fish, we can use trigonometry. Specifically, we'll use the tangent function, which is defined as the ratio of the opposite side (the depth of the fish) to the adjacent side (the horizontal distance to the fish).

Given:

• Depth of the fish (opposite side) = 109 feet
• Horizontal distance to the fish (adjacent side) = 230 feet

We can find the angle \( \theta \) using the following relationship:

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{109}{230} \]

To find \( \theta \), we take the arctangent (inverse tangent) of the ratio:

\[ \theta = \tan^{-1}\left(\frac{109}{230}\right) \]

Now, calculating the value:

\[ \frac{109}{230} \approx 0.4739 \]

Using a calculator to find the arctangent:

\[ \theta \approx \tan^{-1}(0.4739) \approx 25.2^\circ \]

Thus, the approximate angle Demetry will need to drop their fishing line to catch the fish is:

\[ \boxed{25.2^\circ} \]