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Calculus/related rates

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Expert: - 5/17/2010


Question
A balloon, leaving the ground 60 ft.  from an observer, rises vertically at the rate of 10 ft per second.  How fast is the balloon receding from the observer after 8 sec?

Answer
The base of the triangle involves where the baloon lifted off from and where the observer was standing.  This side is said to be 60' and will be called A.

The baloon rises at 10ft/sec, so after 8 sec, it is 80' off the ground.  This is the vertical side of the triangle and will be called B.

The hypotenuse of the triangle, called C, follows the equaion A² + B² = C²
where A, B, and C are all functions of t.

Thus, we can differentiate with respect to t and get 2AA' + 2BB' = 2CC'.
Since A is 60' and is a constant, A' is 0, so that term can be dropped.
Since both sides are multiplied by 2, that can be cancelled.

This leaves B*B' = C*C'.

Now B=80, so that can be put in.
Further, it is known that C = √(A²+B²), so C can be found and put in.
Lastly, it is rising at 10 ft/sec, so B' = 10, and that can be put in.

The only thing not found is what is being looked for, and that is C'.
Thus, since the rest are known, C' = B*B'/C.

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