Sec 4.6 Rates and Related Rates

Ex 1) Water is pumped up a straight cylindrical pipe with a diameter of 7 inches.

The pump moves 3 cubic feet of water a minute.

At what speed will the water move through the pipe?

Ex 2) A voltage, V, across a resistance, R, generates a current, I, where

V is measured in volts, R is measured in ohms, I is measured in amps.

If a constant voltage of 9 volts is put across a resistance that is changing at a rate of 0.2 ohms per second when the resistance is 5 ohms, then at what rate is the current changing at that instant?

Ex 3) Water is being pumped into a hemispherical bowl with radius 20 ft.

3 minutes into the process of filling the bowl, the depth of the water in the bowl is 5 ft. and is increasing at a rate of 1.25 ft/min.

How fast is the diameter of the pool increasing 3 minutes into the filling process?

Ex 4) We are filming a watermelon being dropped from a 150 ft. tower. We are on level ground 10 ft. from the point of impact.

The camera is on a 5 ft. tripod.

The height of the watermelon is given by:

where t is in seconds and h(t) is in ft.

a) Find the function that relates the angle of inclination of the camera to the height of the watermelon.

b) What is the relationship between the rate at which the watermelon is falling and the rate at which the angle of inclination of the camera is changing?

c) At time 3 seconds after the melon is dropped what is its height off the ground?

d) At time 3 seconds after the melon is dropped what is the velocity of the melon?

e) At time 3 seconds after the melon is dropped what is the angle of inclination of the camera.

f) At time 3 seconds after the melon is dropped at what rate is the angle of inclination of the camera changing?

Ex 5) Two trains leave the station at 12 noon.

Train 1 heads east a 50 mph.

Train 2 heads south of west at 65 mph.

At what rate is the distance between the trains changing at 2pm?

Recall the law of cosines:

Ex 6) Water is being pumped up through the pipe drawn below at a constant rate of A .

Draw a graph of the height of the water in the pipe versus time.

Ex 7) Given position data, describe the velocity and acceleration of this object over the time span of the given data.

Time t (sec)	0	5	10	15	20	25	30	35	40	45
Position (ft)	0	23	45	66	86	105	122	137	150	160

Ex 8) X, Y and Z are functions of t.

a, b and c are constants.

Differentiate (implicitly) the following relationship with respect to t.