Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet (Page 20 of 41) in pdf

Worksheet # 15: Related Rates

1. Let a and b denote the length in meters of the two legs of a right triangle. At time t = 0, a = 20 and

b = 20. If a is decreasing at a constant rate of 2 meters per second and b is increasing at a constant

rate of 3 meters per second. Find the rate of change of the area of the triangle at time t = 5 seconds.

2. A person 6 feet tall walks along a straight path at a rate of 4 feet per second away from a streetlight

that is 15 feet above the ground. Assume that at time t = 0 the person is touching the streetlight.

(a) Draw a picture to represent the situation.

(b) Find an equation that relates the length of the person’s shadow to the person’s position (relative

to the streetlight).

(d) Find how fast is the tip of the person’s shadow is moving when t = 4.

(e) Does the precise time make a diﬀerence in these calculations?

3. A spherical snow ball is melting. The rate of change of the surface area of the snow ball is constant

and equal to

7 square centimeters per hour. Find the rate of change of the radius of the snow ball

when r = 5 centimeters.

4. The height of a cylinder is a linear function of its radius (i.e. h = ar + b for some a, b constants). The

height increases twice as fast as the radius r and

is constant. At time t = 1 seconds the radius is

r = 1 feet, the height is h = 3 feet and the rate of change of the volume is 16π cubic feet/second.

(a) Find an equation to relate the height and radius of the cylinder.

(b) Express the volume as a function of the radius.

5. A water tank is shaped like a cone with the vertex pointing down. The height of the tank is 5 meters

and diameter of the base is 2 meters. At time t = 0 the tank is full and starts to be emptied. After

3 minutes the height of the water is 4 meters and it is decreasing at a rate of 0.5 meters per minute.

At this instant, ﬁnd the rate of change of the volume of the water in the tank. What are the units for

your answer? Recall that the volume of a right-circular cone whose base has radius r and of height h

is given by V =

πr

6. A plane ﬂies at an altitude of 5000 meters and a speed of 360 kilometers per hour. The plane is ﬂying

in a straight line and passes directly over an observer.

(a) Sketch a diagram that summarizes the information in the problem.

(b) Find the angle of elevation 2 minutes after the plane passes over the observer.

7. A car moves at 50 miles per hour on a straight road. A house is 2 miles away from the road. What is

the rate of change in the angle between the house and the car and the house and the road when the

car passes the house.

8. A car moves along a road that is shaped like the parabola y = x

. At what point on the parabola are

the rates of change for the x and y coordinates equal?

9. Let f (x) =

and h(x) =

1 + x

1 + f (x)

(a) Find f (x).

(b) Use the previous result to ﬁnd h (x).

x (1).

Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 20

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