Accelerated Calculus
Name: _____________________
Optimization and Related Rates Problems
Oct. 26/27
Directions: Complete each of the following optimization and related rates problems on your own
paper. At least two of these problems will be used on our next test.
1. The minute hand on a watch is 8mm long and the hour hand is 4mm long. How fast is the
distance between the tips of the hands changing at one o’clock?
2. A flood lamp is installed on the ground 200 feet from a vertical wall. A six foot tall man is
walking towards the wall at the rate of 30 feet per second. How fast is the tip of his shadow
moving down the wall when he is 50 feet from the wall?
3. A common volume for small, cylindrical, juice cans is 6 ounces (168 cm3). The cost to
produce such a can is related to the how much metal is needed to make the can. Find the
optimal dimensions that minimize cost for a 6 oz can provided
a. the metal used is of a uniform thickness.
b. the metal used for the top and bottom of the can must be reinforced and costs
twice as much to manufacture as the metal for the sides of the can.
x
4. What point on the curve y  e is closest to the origin? What point is closest to the line
y x?
5. A highway must be constructed to connect town A with town B. There is an existing
roadway that can be upgraded 25 miles north of the line connecting the two towns. The cost
of upgrading the existing roadway is $200,000 per mile, whereas the cost of constructing any
new highway is $300,000 per mile. Describe the path of roadway that will minimize the cost
of construction.
Accelerated Calculus
Optimization and Related Rates Problems
Name: _____________________
Oct. 26/27
6. Water is leaking from an inverted conical tank at a rate of 10,000 cm 3 / min while water is
also being pumped into the tank at a different rate. At any depth, the cone of water has a ratio
of h:r of 3:2. The tank has a height of 6 meters, is painted red, and cost $34,000 to
manufacture in 2002. At noon, the water level is observed to be rising by 20 cm / min while
the height of water is measured to be 2 meters. What is the rate at which water is being
pumped into the tank?
7. Two sides of a triangle have lengths 12m and 15m. The angle between these sides is
increasing at a rate of 2 / min . How fast is the length of the third side increasing when the
angle between the sides of fixed length is 60 . How fast is the area changing?
8. Consider a circular lake with radius 2 miles. You need to arrive at a point diametrically
opposite as quickly as possible. Your walking speed is 4 mi/hr and your rowing speed is 2
mi/hr. What is the quickest route?
9. At noon, Ship A is 100 km west of Ship B. Ship A is sailing south at 35 km/hr and Ship B is
sailing north at 25 km/hr. How fast is the distance between the ships changing at 4:00 pm the
same day?
10. A baseball diamond is a square whose sides are 90ft long. Suppose that a player running
from second base to third base has a speed of 30ft/s.
a. At what rate is the player’s distance from home plate changing when he is 20 ft
from 3rd base?
b. How fast is the angle   , whose vertex is at home plate, changing when the
player is 20 ft from 3rd base?