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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?

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?

Grade:Upto college level

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
6 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
askIITians Faculty

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