Name:
Review for 3.4-3.5 on Optimization and Related Rates
1. Find two numbers, x and y, whose sum is 24 for which the sum of their squares is a minimum.
2. What is the smallest perimeter possible for a rectangle whose area is 30 in2?
3. An open box is made from a piece of metal 20 by 32 inches by cutting out squares of equal size from
the corners and bending up the sides. What size square should be cut out to create a box with greatest
volume?
4. A rectangular area is fenced in along a river. What is the maximum area that can be create with 70
feet of fencing, knowing that you do not need fencing against the river?
5. Find two numbers whose sum is 100, for which their product is a maximum.
6. A circle is increasing at a rate of 5 cm2/sec. How fast is the radius increasing when the radius is 15cm?
7. A balloon is decreasing at a rate of 5 cm3/min. At what rate is the radius decreasing when the radius is
6 cm? Volume of a sphere: V 
4 3
r
3
8. A 17 foot ladder is leaning against a vertical wall with its other end on the ground. The top end of the
ladder is sliding down the wall. When the top end is 8 feet from the ground it is sliding down at 2 ft/sec.
How fast is the bottom moving away from the wall at this instant?
9. A 6 foot tall person is walking away from a 40 foot tall lamppost at a rate of 5 ft/sec. At what rate is
the length of the person’s shadow changing when the person is 15 feet from the lamppost?
10. An observer stands 560 ft away from a launch pad to observe a rocket launch. The rocket blasts off
and maintains a velocity of 900ft/sec. How fast is the observer to rocket distance changing when the
rocket is 1920 ft from the ground?