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Grade Upto college level Algebra

Gas is escaping from a spherical balloon at the rate of 900 cm3/sec. How fast is the surface area, radius of the balloon shrinking when the radius of the balloon is 30cm?

Manvendra Singh chahar

12 Years agoGrade Upto college level

1 Answer

Jitender Singh

12 Years ago

Ans:

Let radius of balloon to be ‘r’

Volume ‘V’:

$V = \frac{4}{3}\pi r^{3}$

$\frac{\partial V}{\partial t} = 4\pi r^{2}.\frac{\partial r}{\partial t} = 900 \frac{cm^{3}}{sec}$

Surface Area ‘S’:

$S = 4\pi r^{2}$

$\frac{\partial S}{\partial t} = 8\pi r.\frac{\partial r}{\partial t} = \frac{2}{r}.4\pi r^{2}.\frac{\partial r}{\partial t}$

$\frac{\partial S}{\partial t} = \frac{2}{30}.4\pi (r)^{2}.\frac{\partial r}{\partial t} = \frac{1}{15}.900 = 60\frac{cm^{3}}{sec}$

$8\pi r.\frac{\partial r}{\partial t} = 60$

$\frac{\partial r}{\partial t} = \frac{60}{8\pi .30} = \frac{1}{4\pi }\frac{cm}{sec}$

Thanks & Regards

Jitender Singh

IIT Delhi

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