Related Rates Problems

Related Rates ProblemsDo Now: How fast is the Circumference of a circle changing compared to the rate of change of its radius?

Problem #1

•How fast is the area of a square increasing at the instant when the sides are 4 feet long if the sides increase at a rate of 1.5 ft/min?

Solution #1

• (units?)

Problem #2

• Suppose . At the instant when x=1 and y=3, x is decreasing at 2 units/sec and y is increasing at 2 units/sec. How fast is z changing at this instant? Increasing or decreasing?

Solution #2

• units?• Units/sec

Problem #3

• Let be the acute angle of a right triangle with legs of length x and y. At a certain instant, x=4 and is increasing at 2 units/second, while y=3 and is decreasing at ½ units/sec. How fast is changing at that instant? Is it increasing or decreasing?

Solution #3

Problem #4

•Assume that oil spilled from a ruptured tanker spreads in a circular pattern whose radius increases at a constant rate of 2 ft/sec. How fast is the area of the spill increasing when the radius of the spill is 60 ft?

Solution #4

Problem #5

• Suppose that a spherical balloon is inflated at the rate of 10 cubic centimeters per minute. How fast is the radius of the balloon increasing when the radius is 5 centimeters?

Solution #5

Problem #6

•One end of a 13-foot ladder is on the floor and the other rests on a vertical wall. If the bottom end is drawn away from the wall at 3 feet per second, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 5 feet from the wall?

Solution #6

• (Where did I get 12?)

Problem #7

•One end of a 13-foot ladder is on the floor and the other rests on a vertical wall. If the bottom end is drawn away from the wall at 3 feet per second, how fast is the angle of elevation of the ladder changing when the bottom of the ladder is 5 feet from the wall?

Solution #7

Problem #8

• A baseball diamond is a square whose sides are 90 ft long. Suppose that a player running from second base to third base has a speed of 30 ft/sec at the instant when he is 20 feet from third base. At what rate is the player’s distance from home plate changing at that instant?

Solution #8

• At this instant

•

Problem #9

•A rocket is rising vertically at 880 ft/sec. At the instant when the rocket is 4000 feet high, how fast must a camera’s elevation angle change to keep the rocket in sight if the camera is 3000 ft away from the launching pad?

Solution #9

• .• At this moment, the hypotenuse is .