MATH-2450 HANDOUT 5 (Lesson 11)
1.
A ladder 20 feet long leans against a vertical building. If the bottom of the ladder slides away from the
building horizontally at a rate of 3 ft/sec, how fast is the ladder sliding down the building when the top of
the ladder is 8 feet from the ground?
2.
Gas is being pumped into a spherical balloon at a rate of 5 ft 3/min. If the pressure is constant, find the rate
at which the radius is changing when the diameter is 18 inches.
3.
A light is at the top of a 16-foot pole. A boy 5 feet tall walks away from the pole at a rate of 4 ft/sec. At
what rate is the tip of his shadow moving when he is 18 feet from the pole? At what rate is the length of
his shadow increasing?
4.
A man on a dock is pulling in a boat by means of a rope attached to the bow of the boat 1 foot above water
level and passing through a simple pulley located on the dock 8 feet above water level. If he pulls in the
rope at a rate of 2 ft/sec, how fast is the boat approaching the dock when the bow of the boat is 25 feet
from a point on the water directly below the pulley?
5.
As sand leaks out of a hole in a container, it forms a conical pile whose altitude is always the same as its
radius. If the height of the pile is increasing at a rate of 6 in/min, find the rate at which the sand is leaking
out when the altitude is 10 inches.
6.
A boy flying a kite pays out string at a rate of 2 ft/sec as the kite moves horizontally at an altitude of 100
feet. Assuming there is no sag in the string, find the rate at which the kite is moving when 125 feet of
string have been paid out.
7.
A point P(x, y) moves on the graph of the equation y = x 3 + x 2 + 1, the abscissa is changing at a rate of 2
units per second. How fast is the ordinate changing at the point (1, 3)? Note: the abscissa is the xcoordinate of the point and the ordinate is the y-coordinate of the point.
8.
The ends of a water trough 8 feet long are equilateral triangles whose sides are 2 feet long. If the water is
being pumped into the trough at a rate of 5 ft 3/min, find the rate at which the water level is rising when the
depth is 8 inches.
9.
The top of a silo has the shape of a hemisphere of diameter 10 feet. If it is coated uniformly with a layer of
ice, and if the thickness is decreasing at a rate of 1/2 inch per hour, how fast is the volume of the ice
changing when the ice is 2 inches thick?
10.
The apparent power Pa of an electric circuit whose power is P and whose impedance phase angle is θ is
given by Pa = P sec θ. Given that P is constant at 12 W and θ is changing at the rate of 0.5 rad/min, find
the rate of change of Pa when θ = 60.
11.
An airplane is flying at an altitude of 4000 feet toward an observer on the ground. The plane is flying at a
rate of 500 ft/sec. How fast is the angle of elevation from the observer to the plane changing when the
plane is directly over the observer?
12.
A missile is fired vertically from a point that is 5 miles from a tracking station. For the first 20 seconds of
flight, its angle of elevation θ changes at a constant rate of 2 per second. Find the velocity of the missile
when the angle of elevation is 30.
13.
A balloon is released and rises vertically at a rate of 10 meters/second. An observer is located 30 meters
from the point where the balloon is released. How fast is the angle of elevation from the observer to the
balloon changing 4 seconds after the balloon is released?
14.
A searchlight located 1/8 mile from the nearest point P on a straight road is trained on an automobile
traveling on the road at a rate of 50 mi/hr. Find the rate at which the searchlight is rotating when the car is
1/4 mile from P.