# A cylinder gets taller at a rate of 3 inches per second, but the radius shrinks at a rate of 1 inch per second. How fast is the volume of the cylinder changing when the height is 20 inches and the radius is 10 inches?

At the instant of interest,

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Volume is decreasing at a rate of

or approximately

Let us setup the following variables:

# {

(r, "Radius of cylinder at time t","(in)"), (h, "Height of cylinder at time t","(in)"), (V, "Volume of cylinder at time t", "(in"^3")"), (t, "time", "(sec)") :} #

Using the standard formula for volume of a cylinder:

Then by the chain rule, we have:

And we are given that

So we have:

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To find how fast the volume of the cylinder is changing when the height is 20 inches and the radius is 10 inches, we use the formula for the volume of a cylinder:

[ V = \pi r^2 h ]

We are given that the height is changing at a rate of 3 inches per second (( \frac{dh}{dt} = 3 )) and the radius is changing at a rate of -1 inch per second (( \frac{dr}{dt} = -1 )). We need to find the rate of change of volume (( \frac{dV}{dt} )) when ( h = 20 ) inches and ( r = 10 ) inches.

Using the given information and the chain rule, we differentiate the volume formula with respect to time:

[ \frac{dV}{dt} = \pi (2rh \frac{dr}{dt} + r^2 \frac{dh}{dt}) ]

Substitute the given values:

[ \frac{dV}{dt} = \pi (2(10)(20)(-1) + (10)^2 (3)) ]

[ \frac{dV}{dt} = \pi (-400 + 300) ]

[ \frac{dV}{dt} = -100\pi ]

Therefore, when the height is 20 inches and the radius is 10 inches, the volume of the cylinder is changing at a rate of ( -100\pi ) cubic inches per second.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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