An angler hooks a trout and reels in his line at 5 in divided by 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 17 ft from the angler.

We can approach this problem using related rates from calculus, given that the situation involves a changing distance and speed.

### Step 1: Define variables
- Let \( x \) be the horizontal distance between the fish and the angler (in feet).
- Let \( y \) be the length of the fishing line (in feet) between the tip of the rod and the fish.
- Let \( h = 11 \) ft represent the height of the tip of the rod above the water.

We are given that:
- The rate at which the line is being reeled in (i.e., the rate of change of \( y \)) is \( \frac{dy}{dt} = -5 \) inches per second. Since \( y \) is in feet, we convert this to feet per second:
  \[
  \frac{dy}{dt} = -\frac{5}{12} \text{ ft/s}.
  \]

The goal is to find the horizontal speed of the fish \( \frac{dx}{dt} \) when the horizontal distance \( x = 17 \) feet.

### Step 2: Relate \( x \) and \( y \)
Using the Pythagorean theorem, the relationship between \( x \), \( y \), and \( h \) is:
\[
y^2 = x^2 + h^2.
\]
Differentiating both sides with respect to time \( t \), we get:
\[
2y \frac{dy}{dt} = 2x \frac{dx}{dt}.
\]
Simplifying this gives:
\[
y \frac{dy}{dt} = x \frac{dx}{dt}.
\]

### Step 3: Find \( y \) when \( x = 17 \) ft
Using the Pythagorean theorem to find \( y \) when \( x = 17 \) ft:
\[
y = \sqrt{x^2 + h^2} = \sqrt{17^2 + 11^2} = \sqrt{289 + 121} = \sqrt{410} \approx 20.25 \text{ ft}.
\]

### Step 4: Solve for \( \frac{dx}{dt} \)
Substitute the known values into the related rates equation:
\[
20.25 \times \left( -\frac{5}{12} \right) = 17 \times \frac{dx}{dt}.
\]
Simplifying:
\[
-8.4375 = 17 \times \frac{dx}{dt},
\]
\[
\frac{dx}{dt} = \frac{-8.4375}{17} \approx -0.496 \text{ ft/s}.
\]

Thus, the horizontal speed of the fish when it is 17 feet from the angler is approximately **0.496 ft/s** (towards the angler).

Would you like more details or have any questions? Here are five related questions:

1. How would the speed change if the fish were 10 ft away instead of 17 ft?
2. What happens to the horizontal speed as the fish gets closer to the angler?
3. How is the Pythagorean theorem applied in related rates problems like this one?
4. Can the formula for related rates be used for other similar real-world problems?
5. What happens to the tension in the line as the fish gets closer?

**Tip:** Always convert units consistently when working with related rates to avoid errors!

An angler hooks a trout and reels in his line at 5 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 17 ft from the angler.

This calculus problem involves using related rates and the Pythagorean theorem to calculate the horizontal speed of a fish being reeled in at 5 in/s when it is 17 feet from the angler. Key concepts include differentiation and unit conversion, making it suitable for high school students studying calculus.

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