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Calculus/related rates- Thanks for your help!!

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Expert: - 1/2/2009


Question
Let V be the volume of a circular cylinder having a height h and a radius r, and assume h and r vary with time. At a certain instant, the height is 6in and is increasing at 1 in/s, while the radius is 10in and decreasing at 1 in/s. How fast is the colume changing at that instant and is the volume increasing or decreasing?

Answer
To do this, three equations are needed:
The volume of a cyllinder, the power rule, and product rule.

The volume of a circular cyllinder is πr²h.
The variables h and r are really h(t) and r(t).

The power rule is for differentiating f(x)g(x).
The derivative is f'(x)g(x) + f(x)g'(x).

The chain rule is for a function to a power, f^n(x).
The derivative is nf^(n-1)(x).

Differentiate the volume with respect to t.
You have a product rule with h(t) and r(t).
You also have a function to a power with the r²(t).

We may not know what t is, but that doesn't matter.
We know that h(t)=6, h'(t)=1, r(t)=10, and r'(t)=1.

The derivative of πr²(t)h(t) is π[2r(t)r'(t)h(t) + r²(t)h'(t)].
As was stated right before this, all of these values are known
{ r(t), h(t), r'(t), and h'(t) }.

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