009A Sample Final 1, Problem 5
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A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing
when 50 (meters) of the string has been let out?
| Foundations: |
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Solution:
| Step 1: |
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| Insert diagram. |
| From the diagram, we have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 30^{2}+h^{2}=s^{2}} by the Pythagorean Theorem. |
| Taking derivatives, we get |
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| Step 2: |
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| If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle s=50} , then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle h={\sqrt {50^{2}-30^{2}}}=40} . |
| So, we have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2(40)6=2(50)s'} . |
| Solving for , we get Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle s'={\frac {24}{5}}} m/s. |
| Final Answer: |
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle s'={\frac {24}{5}}} m/s |