Rate problems in calculus

This is a collection of study problems related to rates of change.

## Example

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a coned-shaped paper drinking cup is to be made to hold 27cu. cm. of water. Find the height and radius of the cup that will use the smallest amount of paper.

## Solution

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First, we have to find a formula which relates the volume of the container, and the height. One possible formula is

V = π r 2 h . {\displaystyle V=\pi r^{2}h\;\,.}

We can put in the time dependence explicitly:

V ( t ) = π [ r ( t ) ] 2 h ( t ) . {\displaystyle V(t)=\pi [r(t)]^{2}h(t)\;\,.}

So, differentiating with respect to time (using the product rule), we get

d V ( t ) d t = π [ r ( t ) ] 2 d h ( t ) d t + 2 π r h d r ( t ) d t {\displaystyle {\frac {dV(t)}{dt}}=\pi [r(t)]^{2}{\frac {dh(t)}{dt}}+2\pi rh{\frac {dr(t)}{dt}}}

Now, one of the crucial observations here is that the radius of the container is not changing. So we know that

d r ( t ) d t = 0 {\displaystyle {\frac {dr(t)}{dt}}=0}

Hence,

d V ( t ) d t = π [ r ( t ) ] 2 d h ( t ) d t {\displaystyle {\frac {dV(t)}{dt}}=\pi [r(t)]^{2}{\frac {dh(t)}{dt}}}

Substituting in all our quantities, we can get the answer!

## Questions

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* A ladder 5 m long is falling down along the side of a wall without slipping. When the top is 3 m from the ground, how fast is the end of the ladder moving away from the wall?

* A cylindrical cone is filled with water at a rate of 2 cm3/s. When the height of the water is 10 cm, both the diameter and the height of the water is expanding at a rate of 1 cm/s. What is the slope of the cone?