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This problem is a classic related rates problem in calculus. Let's break it down step by step.

### Given:
- The child flies a kite at a constant height of 90 feet.
- The kite is being blown horizontally away from the child at a rate of \( \frac{dx}{dt} = 30 \, \text{ft/sec} \), where \( x \) is the horizontal distance between the child and the kite.
- We are asked to find the rate at which the child must let out the string, i.e., \( \frac{ds}{dt} \), when the distance between the child and the kite is 150 feet, where \( s \) is the length of the string.

### Step 1: Set up the relationship between the variables
We can use the Pythagorean theorem to relate the height of the kite, the horizontal distance, and the length of the string:
\[
s^2 = x^2 + h^2
\]
where:
- \( s \) is the length of the string,
- \( x \) is the horizontal distance,
- \( h \) is the height of the kite (which is constant at 90 feet).

### Step 2: Differentiate with respect to time \( t \)
Since \( h \) is constant, we can differentiate both sides of the equation with respect to time \( t \):
\[
2s \frac{ds}{dt} = 2x \frac{dx}{dt}
\]
Simplifying this:
\[
s \frac{ds}{dt} = x \frac{dx}{dt}
\]

### Step 3: Plug in the known values
We are given that when the kite is 150 feet away from the child, we need to find \( \frac{ds}{dt} \). First, solve for \( s \) when the kite is 150 feet away:
\[
s = 150 \, \text{ft}
\]
The height of the kite is 90 feet, so the horizontal distance \( x \) is:
\[
x^2 + 90^2 = 150^2
\]
\[
x^2 + 8100 = 22500
\]
\[
x^2 = 22500 - 8100 = 14400
\]
\[
x = \sqrt{14400} = 120 \, \text{ft}
\]

Now, substitute the known values into the differentiated equation:
\[
150 \cdot \frac{ds}{dt} = 120 \cdot 30
\]
\[
150 \cdot \frac{ds}{dt} = 3600
\]
\[
\frac{ds}{dt} = \frac{3600}{150} = 24 \, \text{ft/sec}
\]

### Final Answer:
The child must let out the string at a rate of \( 24 \, \text{ft/sec} \) when the kite is 150 feet away.

Let me know if you'd like further clarification or details on this! 

Here are 5 related questions:
1. What would happen if the height of the kite changed over time?
2. How would the problem change if the kite's horizontal speed varied instead of being constant?
3. How would you apply related rates if the wind speed increased?
4. What happens if the child is walking towards the kite while letting out the string?
5. How would you solve this problem if the horizontal distance was given in meters instead of feet?

**Tip:** Always identify which quantities are constant and which are changing when solving related rates problems.

A child flies a kite at a height of 90 ft, the wind carrying the kite horizontally away from the child at a rate of 30 ft/sec. How fast must the child let out the string when the kite is 150 ft away from the child?

Solve a related rates problem where a child flies a kite at a height of 90 ft, with wind moving it horizontally at 30 ft/sec. Learn how to calculate the rate at which the string must be let out when the kite is 150 ft away.

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