Find an unknown rate of change by relating it to other variables whose values and rate of change at time
Solving strategy
(Matheno)
- Draw a picture of the physical situation
- Write an equation that relates the quantities of interest.
- Be sure to label as a variable any value that changes as the situation progresses; don’t substitute a number for it yet.
- To develop equation, make use of:
- a simple geometric fact (like the relation between a circle’s area and its radius, or the relation between the volume of a cone and its base-radius and height); or
- a trigonometric function (like
= opposite/adjacent); or - the Pythagorean theorem; or
- similar triangles.
- Take the derivative with respect to time of both sides of your equation.
- Solve for the quantity you’re after.
Example 1: Rate of change in volume
A spherical ballon will pop if its radius is growing at rate larger than 1cm/s. What is the maximum rate of increasing its volume when its radius is 5cm?
We know:
Find:
Example 2: Rate of change in height
A water tank is shaped like an upside down cone that is 4 m high and 2 m in radius. Water leaks out at a rate of
. At what rate is the height of the water dropping when the water is 0.7m deep? We need to find the rate of height by time
, so we should differentiate the water volume by the height. Convert radius to height:
Example 3: Rate of change in a hypotenuse
A person stands 15 meters east of an intersection and watches a car driving towards the intersection from the north at 1 meter per second. At a certain instant, the car is 8 meters from the intersection. What is the rate of change of the distance between the car and the person at that instant (in meters per second)?
We know
Find