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Question

The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at the rate of 0.2 cm/min. The rate of change of the volume of the cylinder, in cm3/min, when the radius is 2 cm and the height is 3 cm is
(The negative sign(-) indicates that volume decreases)


A
2π5 
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B
8π5 
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C
3π5
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D
2π5
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Solution

The correct option is C 2π5
Let r and h be the radius and height of the cylinder respectively.
Given, r=2 cm, h=3 cm
drdt=0.1 cm/min
dhdt=0.2 cm/min

V=πr2h
Differentiating both sides
dVdt=π(r2dhdt+2rdrdth)      =πr(rdhdt+2drdth)
      =πr(r(0.2)+2h(0.1))      =πr5(r+h)
Thus, when r=2 and h=3,
dVdt=π(2)5(2+3)=2π5cm3/min

Mathematics

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