Rate of increase

Calculus Level 3

A balloon in the form of a sphere is being filled with water and it is observed that its surface area increases at the rate of 44%. The volume of the balloon is increasing at the rate of?

50 % 44 % 72.8 % Data Insufficient

Sajjan Barnwal by Sajjan Barnwal , 30, India

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1 solution

Rahul Singh
Aug 21, 2015

We have, v= 4 3 \frac{4}{3} pi r 3 r^{3}

or d V d t \frac{dV}{dt} =4pi r 2 r^{2} d r d t \frac{dr}{dt} ......(i)

S=4pi r 2 r^{2}

or d S d t \frac{dS}{dt} =8pi*r d r d t \frac{dr}{dt}

or d r d t \frac{dr}{dt} = 44 8 p i ∗ r \frac{44}{8pi*r} [since, d S d t \frac{dS}{dt} =44]

Substititing the value of d r d t \frac{dr}{dt} in equation (i), we get,

d V d t \frac{dV}{dt} =22r, which is dependent on 'r'.

Legend:

V=Volume of the balloon

S=Surface area of the balloon

r=Radius of the balloon

t=Time

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