Stewart_Calc_7ET ch03sec09
MULTIPLE CHOICE
1. Two cars start moving from the same point. One travels south at mi/h and the other
travels west at mi/h. At what rate is the distance between the cars increasing 2 hours
later? Round the result to the nearest hundredth.
a.
b.
c.
d.
e.
2. A plane flying horizontally at an altitude of 1 mi and a speed of mi/h passes directly
over a radar station. Find the rate at which the distance from the plane to the station is
increasing when it is 2 mi away from the station.
a.
b.
c.
d.
e.
3. Two sides of a triangle are m and m in length and the angle between them is increasing
at a rate of rad/s. Find the rate at which the area of the triangle is increasing when the
angle between the sides of fixed length is .
a.
b.
c.
d.
e.
4. Gravel is being dumped from a conveyor belt at a rate of ft/min and its coarseness is
such that it forms a pile in the shape of a cone whose base diameter and height are always
equal. How fast is the height of the pile increasing when the pile is ft high? Round the
result to the nearest hundredth.
a.
b.
c.
d.
e.
5. The top of a ladder slides down a vertical wall at a rate of m/s . At the moment when
the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2
m/s . How long is the ladder?
a.
b.
c.
d.
e.
None of these
6. If two resistors with resistances and are connected in parallel, as in the figure, then
the total resistance measured in ohms (), is given by
.
If and are increasing at rates of and respectively, how fast is
changing when and ?
Round the result to the nearest thousandth.
a.
b.
c.
d.
e.
NUMERIC RESPONSE
1. Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley (see the
figure below). The point Q is on the floor 12 ft directly beneath and between the carts. Cart
A is being pulled away from Q at a speed of ft/s. How fast is cart B moving toward Q at
the instant when cart A is 5 ft from Q?
2. If a snowball melts so that its surface area decreases at a rate of , find the rate at
which the diameter decreases when the diameter is cm.
3. The altitude of a triangle is increasing at a rate of while the area of the triangle is
increasing at a rate of . At what rate is the base of the triangle changing when the
altitude is 10 cm and the area is .
4. A water trough is 20 m long and a cross-section has the shape of an isosceles trapezoid that
is 20 cm wide at the bottom, 60 cm wide at the top, and has height 50 cm. If the trough is
being filled with water at the rate of , how fast is the water level rising when the
water is cm deep? Round the result to the nearest hundredth.
5. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a
pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a
rate of 1 m/s how fast is the boat approaching the dock when it is m from the dock?
Round the result to the nearest hundredth if necessary.
6. The volume of a cube is increasing at a rate of . How fast is the surface area
increasing when the length of an edge is .
7. A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first
base with a speed of ft/s. At what rate is his distance from second base decreasing when
he is halfway to first base? Round the result to the nearest hundredth.
SHORT ANSWER
1. The top of a ladder leaning against a wall is 8 ft above the ground. The slope of the ladder
with respect to the ground is –4. What is the length of the ladder?