Math 1100
Lecture Notes
30 September, 2011
Steps to Solve Optimization Problems
1. Draw a picture. Label it with variables. Decide what to solve for.
2. Write an equation relating your variables. (Use only general information!)
3. Simplify the equation as much as possible using only general information.
4. Differentiate with respect to time.
5. Plug in particular or general information.
6. Solve for the variable determined in step 1.
Example 1
The radius of a sphere is increasing at the rate of 4 cm/s. How fast is the surface area
increasing when the radius is 2 cm?
(Remember that the equation for surface area of a sphere is A = 4πr2 .)
Example 2
Sand falls at a rate of 5 ft3 /min onto a conical pile. The diameter of the base is always equal
to the height of the pile. At what rate is the height increasing when the pile is 10 ft high?
Example 3
A hot air balloon has a velocity of 50 ft/min and is flying at a constant height of 500 ft. An
observer on the ground is watching the balloon approach. How fast is the distance between
the observer and the balloon changing when the balloon is 1000 ft from the observer?
Example 4
Two cars are approaching an intersection on roads that are perpendicular to each other. Car
A is north of the intersection and traveling south at 40 mph. Car B is east of the intersection
traveling west at 55 mph. How fast is the distance between the cars changing when car A is
15 miles from the intersection and car B is 8 miles from the intersection?
Example 5
A baseball diamond is a square with side length 90 ft. A batter hits a ball and runs toward
first base at a speed of 24 ft/s.
(a) At what rate is his distance from second base decreasing when he is halfway to first base?
(b) At what rate is his distance from third base increasing at the same moment?
Homework: Section 11.4, questions 14,16,23,24,30,31,32,33,36