By Pythagorean theorem and differential calculus, the child must let out the string when the kite is 150 feet away at a rate of change of 25.725 feet per second.
How to determine the rate of change of the string
The geometric representation of this question is based on the Pythagorean theorem:
l² = x² + y² (1)
Where:
- x - Horizontal distance of the kite from the kite, in feet.
- y - Height of the kine, in feet.
- l - Length of the string, in feet.
The rate of change of the length of the string is obtained by differential calculus:
[tex]2 \cdot l \cdot \dot l = 2\cdot x \cdot \dot x[/tex]
[tex]\dot l = \frac{x\cdot \dot x}{l}[/tex]
[tex]\dot l = \frac{x\cdot \dot x}{\sqrt{x^{2}+y^{2}}}[/tex] (2)
If we know that x = 150 ft, y = 90 ft and [tex]\dot x = 30\,\frac{ft}{s}[/tex], then the rate of change of the length of the string is:
[tex]\dot l = \frac{(150\,ft) \cdot \left(30 \,\frac{ft}{s} \right)}{\sqrt{(150\,ft)^{2}+(90\,ft)^{2}}}[/tex]
[tex]\dot l \approx 25.725\,\frac{ft}{s}[/tex]
By Pythagorean theorem and differential calculus, the child must let out the string when the kite is 150 feet away at a rate of change of 25.725 feet per second.
To learn more on rates of change: https://brainly.com/question/13103052
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