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AP Calculus AB AP Review 2.
- The volume of a cone of a radius r and height h is given by
V = π r h. If the radius
and the height both increase at a constant rate of ½ cm/s, at what rate, in cm 3 /s, is the volume increasing when the height is 9 cm and the radius is 6 cm. a. π/ b. 10 π c. 24 π d. 54 π e. 108 π
- The sides of the rectangle shown in the figure increase in such a way that dz / dt = 1 and dx / dt = 3 dy / dt. At the instant when x = 4 and y = 3, what is the value of dx / dt? a. 1/ b. 1 c. 2
d. 5
e. 5
- The radius r of a sphere is increasing at a constant rate of 0.04 centimeters per
second. (Note: The volume of a sphere with radius r is
V = π r )
a. At the time when the radius of the sphere is 10 cm, what is the rate of increase of its volume?
b. At the time when the volume of the sphere is 36π cubic cm, what is the rate of increase of the area of a cross section through the center of the sphere?
c. At the time when the volume and the radius of the sphere are increasing at the same numerical rate, what is the radius?
- In the figure to the right, line
is tangent to the graph of
2
y
x
= at point P , with
coordinates (^2)
w ,
w
, where
w > 0. Point Q has
coordinates ( w ,0). Line crosses the x-axis at point R , with coordinates ( k ,0). a. Find the value of k when w = 3.
b. For all w > 0, find k in terms of w.
c. Suppose that w is increasing at the constant rate of 7 units per second. When w = 5 what is the rate of change of k with respect to time?
d. Suppose that w is increasing at the constant rate of 7 units per second. When w = 5 what is the rate of change of the area of ∆PQR with respect to time? Determine whether the area is increasing or decreasing at this instant.
