Rate problems in calculus

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

Example

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

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

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

We can put in the time dependence explicitly:

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

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

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

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

${\frac {dr(t)}{dt}}=0$

Hence,

${\frac {dV(t)}{dt}}=\pi [r(t)]^{2}{\frac {dh(t)}{dt}}$

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

Questions

• 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?