A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 /min and the thickness of the ice-cream layer decreases at the rate of cm/min. Th

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Solution

Q.
A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 ${cm}^{3}$ /min and the thickness of the ice-cream layer decreases at the rate of $\frac{1}{4 π}$ cm/min. The surface area (in ${cm}^{2}$ ) of the chocolate ball (without the ice-cream layer) is: [2025]

1 128 $π$

2 225 $π$

3 196 $π$

4 256 $π$

Ans.
(4)

Volume of spherical chocolate,

$V = \frac{4}{3} π r^{3}$

Derivative w.r.t. t, we get

$\frac{d V}{d t} = 4 π r^{2} \frac{d r}{d t}$

$\Rightarrow 81 = 4 π r^{2} \times \frac{1}{4 π}$

$\Rightarrow r^{2} = 81 \Rightarrow r = 9$

$∴$ Surface area of chocolate = $4 π {(r - 1)}^{2}$

= $4 π {(9 - 1)}^{2}$

= $256 π {cm}^{2}$

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