View source for 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
:<math>V = \pi r^2 h \;\,.</math>

We can put in the time dependence explicitly:
:<math>V(t) = \pi [r(t)]^2 h(t) \;\,.</math>

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

:<math>\frac{dV(t)}{dt} = \pi [r(t)]^2 \frac{dh(t)}{dt} + 2\pi r h \frac{dr(t)}{dt}</math>

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

:<math>\frac{dr(t)}{dt} = 0</math>

Hence,
:<math>\frac{dV(t)}{dt} = \pi [r(t)]^2 \frac{dh(t)}{dt}</math>

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 cm<sup>3</sup>/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?
[[Category:Mathematics]]