Mixed Problem Set- Related Rates
1. A conical tank is being filled with water. The tank has height 4 ft and radius 3 ft. If water is being pumped in at a constant rate of 2 cubic inches per minute, find the rate at which the height of the cone changes when the height is 26 inches. Note the difference in units.
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2.A searchlight is positioned 10 meters from a sidewalk. A person is walking along the sidewalk at a constant speed of 2 meters per second. The searchlight rotates so that it shines on the person. Find the rate at which the searchlight rotates when the person is 25 meters from the searchlight.
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3. A person 5 feet tall is walking toward an18 foot pole. A light is positioned at the top of the pole. Find the rate at which the length of the person’s shadow is changing when the person is 30 feet from the pole and walking at a constant speed of 6 feet per second.
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4. The length of a rectangle increases by 3 feet per minute while the width decreases by 2 feet per minute. When the length is 15 feet and the width is 40 feet, what is the rate at which the following changes:
a. area:
b. perimeter:
c. diagonal
5. The volume of a tree is given by whereis the circumference of the tree in meters at
ground level and h is the height of the tree in meters. Both C and h are functions of time t in years.
a. Find a formula for . What does it represent in practical terms?
b. Suppose the circumference grows at a rate of 0.2 meters/year and the height grows at a rate of 4 meters/year. How fast is the volume of the tree growing when the circumference is 5 meters and the height is 22 meters?
6. a. When the radius of a spherical balloon is 10 cm, how fast is the volume of the balloon changing with respect to change in its radius?
b. If the radius of the balloon is increasing by 0.5 cm/sec, at what rate is the air being blown into the balloon when the radius is 6 cm?
7. When hyperventilating, a person breathes in and out very rapidly. A spirogram is a machine that
draws a graph of the volume of air in a person’s lungs as a function of time. During hyperventilation, the person’s spirogram trace might be represented by where V is the volume of air in liters in the lungs at time t minutes.
a. Sketch a graph of one period of this function.
b. What is the rate of flow of air in liters/minute? Sketch a graph of this function.
c. Mark the following on each of the graphs above.
i) the interval when the person is breathing in
(0.005, 0.01)
ii) the interval when the person is breathing out
(0, 0.005)
iii) the time when the rate of flow of air is a maximum when the person is breathing in
0.0075 min