IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

saloni_p

assuming that the volume of the trunk is directly proportional to the cube of its diameter,and that the latter increases uniformly from year to year,show that the rate of increase in the volume when the diameter is equal to 90 cm is 25 times the rate when its 18cm.

iitkgp_bipin

Since volume is directly proportional to the cube of the diameter let :

V = kx³

where V is the volume at any instant , k is any constant

and x is the diameter at any instant

Now differentiate this equation wrt time :

dV/dt = 3kx².dx/dt

Given rate of increase of diameter is constant i.e. dx/dt is constant.

Hence (dV/dt)₂ : (dV/dt)₁ = (x₂/x₁)² = (90/18)² = 25

Hence (dV/dt)₂ = 25.(dV/dt)₁

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