The volume of a spherical hot air ballon expands as the air inside the ballon is heated. The radius of the ballon, in feet, is modeled by a twice-differentiable function r(t), where t is measured in minutes. The table below gives selected values of the rate of change, r'(t) of the radius of the ballon over the time interval 0 ≤ t ≤ 12. The radius of the ballon is 30 feet when t = 5. Note the volume of the sphere of radius r is given by V = (4/3) π r3. Estimate the radius of the ballon when t = 4.8. Find the rate of change of the volume of the ballon with respect to time when t = 5. Show that there must be some time when the rate of change of the radius of the ballon is 3 feet per minute. Estimate the rate of change of r'(t) at t = 9.
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T (minutes)
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0
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2
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5
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7
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11
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12
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r' (t) (feet per minute)
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5.7
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4.0
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2.0
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1.2
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0.6
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0.5
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