Section 2.6 Related Rates

What exactly is a “Related rate”? a “related rate” problem occurs when two or more variables are changing with respect to TIME in a given situation (means that t is the “chief” variable and you therefore differentiate with respect to t)

Procedure in solving these types of problems… pg. 150, gray box entitled “Guidelines For Solving Related-Rate Problems” (copy into your notebook)

TyPE #1: expanding/shrinking object A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 2 ft/sec. When the radius is 5 feet, at what rate is the total area A of the disturbed water changing?

Another example: Air is being left out of a spherical balloon at a rate of 3.5 cm3/min.  Determine the rate at which the radius of the balloon is decreasing when the diameter of the balloon is 20 cm.

AND ONE MORE… Oil spilled from a ruptured tanker spreads out in a circle whose area increases at a constant rate of 6 mi2/hr.  How fast is the radius of the spill increasing when the area is 9 square miles?

TYPE #2: The “slipping ladder” An 8 foot long ladder is leaning against a wall.  The top of the ladder is sliding down the wall at the rate of 2 feet per second.  How fast is the bottom of the ladder moving along the ground at the point in time when the bottom of the ladder is 4 feet from the wall.

ANOTHER EXAMPLE… A 25-foot ladder is leaning against a vertical wall. The floor is slightly slippery and the foot of the ladder slips away from the wall at the rate of 0.2 inches per second. How fast is the top of the ladder sliding down the wall when the top is 20 feet above the floor?

And one more… A 15 foot ladder is resting against the wall.  The bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of .25 ft/sec.  How fast is the top of the ladder moving up the wall 12 seconds after we start pushing?

Type #3: The filling (or unloading) cone/trough A conical water tank with vertex down has a radius of 10 feet at the top and is 24 feet high. If water flows out of the tank at a rate of 20 cubic feet per minute, how fast is the depth of the water decreasing when the water is 16 feet deep?

Another one… Water is flowing into an inverted right circular cone at a rate of 4 cubic inches per minute. The cone is 16 inches tall and its base has a radius of 4 inches. At the moment the water has a depth of 5 inches, how fast is the radius at the surface of the water increasing?

And one more… A tank of water in the shape of a cone is leaking water at a constant rate of 2 cubic feet per hour.  The base radius of the tank is 5 ft and the height of the tank is 14 ft.  (a) At what rate is the depth of the water in the tank changing when the depth of the water is 6 ft? (b) At what rate is the radius of the top of the water in the tank changing when the depth of the water is 6 ft?

TYPE #4: The “pushing” problem A person is pushing a box up a 30 foot long ramp at the rate of 4.5 feet per second. The ramp rises to a height of 7 feet. How fast is the box rising? A package, while being pushed along a conveyor belt “ramp”, is being risen at a rate of 3.5 feet per second. If the conveyor belt is 40 feet long and rises to a height of 5 feet, how fast is the package being pushed horizontally along the belt?