Answer

Verified

420k+ views

Hint: We can write the expression for volume of a balloon and from there we find the rate of change. Volume of a sphere is given by $\dfrac{4}{3}\pi {{r}^{3}}$ . To find Rate of change we use derivatives, so we can write the rate of change of volume with time by differentiating volume with respect to time.

Complete step-by-step answer:

First of all we collect the information which is given in the question which is,

Balloon always remains spherical $\Rightarrow $ Volume of balloon $=\dfrac{4}{3}\pi {{r}^{3}}$

Balloon is being inflated $\Rightarrow \dfrac{dV}{dt}=900c{{m}^{3}}$ which is positive as we are pumping air in and the volume is increasing.

Now we need to find out the rate at which the radius of the balloon increases. For this we will write-

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

Differentiating both sides with respect to time we have,

$\begin{align}

& \dfrac{dV}{dt}=\left( \dfrac{4}{3}\pi \right)\dfrac{d}{dt}({{r}^{3}}) \\

& =\left( \dfrac{4}{3}\pi \right)(3{{r}^{2}})\dfrac{dr}{dt} \\

\end{align}$

On further simplification we have,

$\dfrac{dV}{dt}=4\pi {{r}^{2}}\dfrac{dr}{dt}$

We are given $\dfrac{dV}{dt}=900c{{m}^{3}}$ and we need to find out $\dfrac{dr}{dt}$ when r= 15 cm.

Therefore we write,

$\dfrac{dr}{dt}=\dfrac{1}{4\pi {{r}^{2}}}\dfrac{dV}{dt}$

Substituting r=15 and $\dfrac{dV}{dt}=900c{{m}^{3}}$ we have,

$\dfrac{dr}{dt}=\dfrac{1}{4\pi {{(15)}^{2}}}.900$ cm/s

On simplifying we have,

$\dfrac{dr}{dt}=\dfrac{1}{\pi }cm/s$

Hence, the answer is $\dfrac{1}{\pi }cm/s$ .

Note:

There are a number of things to keep in mind. First of all while differentiating we must take care about with respect to what we are differentiating. If we differentiate with respect to ‘r’ we will not get anything useful. And we should also keep in mind about the units, which will also help us in understanding with respect to what we need to differentiate. We should also keep in mind the sign of rate of change as rate of change may be increasing as well as decreasing. Increasing rate of change will have positive signs and decreasing rate of change will have negative signs.

Complete step-by-step answer:

First of all we collect the information which is given in the question which is,

Balloon always remains spherical $\Rightarrow $ Volume of balloon $=\dfrac{4}{3}\pi {{r}^{3}}$

Balloon is being inflated $\Rightarrow \dfrac{dV}{dt}=900c{{m}^{3}}$ which is positive as we are pumping air in and the volume is increasing.

Now we need to find out the rate at which the radius of the balloon increases. For this we will write-

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

Differentiating both sides with respect to time we have,

$\begin{align}

& \dfrac{dV}{dt}=\left( \dfrac{4}{3}\pi \right)\dfrac{d}{dt}({{r}^{3}}) \\

& =\left( \dfrac{4}{3}\pi \right)(3{{r}^{2}})\dfrac{dr}{dt} \\

\end{align}$

On further simplification we have,

$\dfrac{dV}{dt}=4\pi {{r}^{2}}\dfrac{dr}{dt}$

We are given $\dfrac{dV}{dt}=900c{{m}^{3}}$ and we need to find out $\dfrac{dr}{dt}$ when r= 15 cm.

Therefore we write,

$\dfrac{dr}{dt}=\dfrac{1}{4\pi {{r}^{2}}}\dfrac{dV}{dt}$

Substituting r=15 and $\dfrac{dV}{dt}=900c{{m}^{3}}$ we have,

$\dfrac{dr}{dt}=\dfrac{1}{4\pi {{(15)}^{2}}}.900$ cm/s

On simplifying we have,

$\dfrac{dr}{dt}=\dfrac{1}{\pi }cm/s$

Hence, the answer is $\dfrac{1}{\pi }cm/s$ .

Note:

There are a number of things to keep in mind. First of all while differentiating we must take care about with respect to what we are differentiating. If we differentiate with respect to ‘r’ we will not get anything useful. And we should also keep in mind about the units, which will also help us in understanding with respect to what we need to differentiate. We should also keep in mind the sign of rate of change as rate of change may be increasing as well as decreasing. Increasing rate of change will have positive signs and decreasing rate of change will have negative signs.

Recently Updated Pages

Assertion The resistivity of a semiconductor increases class 13 physics CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

Trending doubts

First census of India was done in A 1871 1881 B 1911 class 12 biology CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How many lightyears away is the sun from the earth class 6 social science CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE