https://file.aiquickdraw.com/mathbot/file/1727627287095uczu3pal.jpg

An angler hooks a trout and reels in his line at 

3 in divided by s3 in/s. 

Assume the tip of the fishing rod is 

11 ft 

above the water and directly above the​ angler, and the fish is pulled horizontally directly toward the angler​ (see figure). Find the horizontal speed of the fish when it is 

16 ft

from the angler.

The scenario involves an angler reeling in a fish, which pulls the fishing line horizontally toward the angler, and we are asked to find the horizontal speed of the fish at a certain moment. The information provided includes:

- The height of the fishing rod (11 ft).
- The horizontal distance of the fish from the angler (16 ft).
- The rate at which the fishing line decreases in length is 3 inches per second (3 in/s).

We can apply related rates to solve this problem. Let's break down the steps:

### Step 1: Define Variables
- Let \( x \) be the horizontal distance between the angler and the fish.
- Let \( L \) be the length of the fishing line (which forms the hypotenuse of the right triangle formed by the fishing rod and the horizontal distance from the angler to the fish).
- The height of the fishing rod is constant at 11 feet, so the vertical side of the right triangle is always 11 ft.

### Step 2: Apply the Pythagorean Theorem
Since the fishing line forms a right triangle with the rod and the horizontal distance, we can use the Pythagorean theorem:
\[
L^2 = x^2 + 11^2
\]
\[
L^2 = x^2 + 121
\]

### Step 3: Differentiate with Respect to Time \( t \)
Differentiate both sides of the equation with respect to time \( t \), using the chain rule:
\[
\frac{d}{dt}(L^2) = \frac{d}{dt}(x^2 + 121)
\]
\[
2L \frac{dL}{dt} = 2x \frac{dx}{dt}
\]

Now, we simplify:
\[
L \frac{dL}{dt} = x \frac{dx}{dt}
\]

### Step 4: Use Given Information
- \( L \) is the length of the fishing line.
- \( \frac{dL}{dt} \) is given as \( -3 \) inches per second (because the line is getting shorter as the fish is pulled in).
- The horizontal distance \( x \) is given as 16 ft.

First, find \( L \) when \( x = 16 \) ft:
\[
L^2 = 16^2 + 11^2 = 256 + 121 = 377
\]
\[
L = \sqrt{377} \approx 19.42 \text{ ft}
\]

Next, substitute \( L \), \( \frac{dL}{dt} = -3 \text{ in/s} \) (note that 3 in/s = \( \frac{3}{12} \) ft/s = 0.25 ft/s), and \( x = 16 \) ft into the related rates equation:
\[
19.42 \times (-0.25) = 16 \times \frac{dx}{dt}
\]
\[
-4.855 = 16 \times \frac{dx}{dt}
\]
\[
\frac{dx}{dt} = \frac{-4.855}{16} \approx -0.303 \text{ ft/s}
\]

### Step 5: Conclusion
The horizontal speed of the fish when it is 16 ft from the angler is approximately \( 0.303 \text{ ft/s} \). Since the fish is moving toward the angler, the speed is positive in the direction of the angler.

Would you like further clarification on any part of the solution?

Here are some related questions:
1. How does the Pythagorean theorem apply in related rates problems?
2. Why does the rate of change of the length of the fishing line affect the fish's speed?
3. What would happen to the fish’s horizontal speed if the fishing line were shorter?
4. How would the problem change if the angler were standing at a different height?
5. What are the units used when converting from inches per second to feet per second?

Tip: When solving related rates problems, always start by writing down the relationship between the variables using a formula like the Pythagorean theorem before differentiating.

An angler hooks a trout and reels in his line at 3 in/s. Assume the tip of the fishing rod is 11 ft above the water and directly above the angler, and the fish is pulled horizontally directly toward the angler. Find the horizontal speed of the fish when it is 16 ft from the angler.

Solve a related rates problem where an angler reels in a fish at 3 in/s. The fishing rod is 11 ft above the water, and the fish is 16 ft away from the angler. Learn how to apply the Pythagorean theorem and differentiate to find the horizontal speed of the fish.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation