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This is a related rates problem involving the movement of a kite.

Let’s define the variables:

- \( h = 80 \) feet is the height of the kite (which remains constant).
- \( x \) is the horizontal distance from the child to the kite, and it changes as the wind carries the kite away at a rate of \( \frac{dx}{dt} = 34 \) ft/sec.
- \( s \) is the length of the string, which changes as the child lets out the string at a rate \( \frac{ds}{dt} \) (this is what we are trying to find).

We are given that when the kite is 170 feet away from the child, \( s = 170 \) feet. The horizontal distance from the child is changing at a rate of 34 ft/sec, and we need to find \( \frac{ds}{dt} \).

Using the Pythagorean theorem, since the kite's height forms a right triangle with the horizontal distance and the string:
\[
s^2 = x^2 + h^2
\]

Differentiate both sides with respect to time \( t \):
\[
2s \frac{ds}{dt} = 2x \frac{dx}{dt}
\]

Simplify:
\[
s \frac{ds}{dt} = x \frac{dx}{dt}
\]

Now, we need \( x \) when \( s = 170 \) feet. From the Pythagorean theorem:
\[
170^2 = x^2 + 80^2
\]
\[
28900 = x^2 + 6400
\]
\[
x^2 = 22500
\]
\[
x = 150 \text{ ft}
\]

Now plug in the values:
\[
170 \frac{ds}{dt} = 150 \times 34
\]
\[
170 \frac{ds}{dt} = 5100
\]
\[
\frac{ds}{dt} = \frac{5100}{170} = 30 \, \text{ft/sec}
\]

Thus, the child must let out the string at a rate of **30 ft/sec** when the kite is 170 feet away from the child.

Would you like more details on any step or have any questions? 

Here are some related questions to deepen your understanding:
1. How would the answer change if the kite was flying higher or lower?
2. What happens if the horizontal velocity of the kite changes?
3. How is the rate of change of the string affected by the height of the kite?
4. Can you apply this method to a situation where the kite's height is not constant?
5. What if the wind moves the kite in two directions: horizontally and vertically?

**Tip**: In related rates problems, always find the relationship between the quantities before differentiating!

A child flies a kite at a height of 80 ft, the wind carrying the kite horizontally away from the child at a rate of 34 ft/sec. How fast must the child let out the string when the kite is 170 ft away from the child?

Solve a related rates problem involving a kite flying at a height of 80 ft and the wind carrying it horizontally at 34 ft/sec. Learn how to use the Pythagorean theorem and differentiation to find the rate at which the string is let out when the kite is 170 ft away from the child.

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