Application of Derivatives 2 Question 1

1. A spherical iron ball of radius 10cm is coated with a layer of ice of uniform thickness that melts at a rate of 50cm3/min. When the thickness of the ice is 5cm, then the rate at which the thickness (in cm/min ) of the ice decreases, is

(2019 Main, 10 April II)

(a) 19π

(b) 118π

(c) 136π

(d) 56π

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Solution:

  1. Let the thickness of layer of ice is xcm, the volume of spherical ball (only ice layer) is

V=43π[(10+x)3103]

On differentiating Eq. (i) w.r.t. ’ t ‘, we get

dVdt=43π(3(10+x)2)dxdt=50

[-ve sign indicate that volume is decreasing as time passes].

4π(10+x)2dxdt=50

At x=5cm

dxdt[4π(10+5)2]=50

dxdt=50225(4π)=19(2π)=118πcm/min

So, the thickness of the ice decreases at the rate of 118πcm/min.



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