A spherical balloon is inflated with gas at a rate of 600 cubic centimeters per

Question

A spherical balloon is inflated with gas at a rate of 600 cubic centimeters per minute (a) Find the rates of change of the radius when r 40 centimeters and r 75 centimeters 40 (b) Explain why the rate of change of the radius of the sphere is not constant even though ava is constant. O The volume only appears constant; it is actually a rational depends on net simply r O The rate of change of the radius is a cubic relationship. The rate of change of the radius is a linear relationship whose slope is O as a function runs parallel to the volume function, which is not linear. All edges of a cube are expanding at a rate of 2 centimeters per second (a) How fast is the surface area changing when each edge is 3 cm2/sec (b) How fast is the surface area changing when each edge is 10 centimeters? cm2/sec At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 12 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 2 feet high? (Hint: The formula for the volume of a cone is v-,%) ft/min

Explanation / Answer

5. V=(4/3) pi r3

dV/dt =(4/3) 3pi r2(dr/dt)

600= 4 pi (40)2(dr/dt)

dr/dt= 0.03 cm/min

At r=75

600= 4 pi (75)2 (dr/dt)

dr/dt =0.008 cm/min

b. dV/dt depends on r2,not simply on r .

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