Question
A balloon which always remains spherical, has a variable diameter $\frac{3}{2} (2\text{x + 1)}.$ Find the rate of change of its volume with respect to x.
✓
Answer
Given: Diameter of the balloon $=\frac{3}{2}(2\text{x}+1)$
$\therefore $ Radius of the balloon $=\frac{3}{4}(2\text{x}+1)$
$\therefore$ Volume of the balloon $=\frac{4}{3}\pi\Big(\frac{3}{4}\pi2\text{x}+1\Big)=\frac{9\pi}{16}(2\text{x}+1)^3$ cu. units
$\therefore$ Rate of change of volume $\text{w.r.t.x}\ = \frac{\text{dV}}{\text{dx}}= \frac{9\pi}{16}.3(2\text{x}+1)^2 .\frac {\text{d}}{\text{dx}}(2\text{x}+\text{x})$
$=\frac{ 27\pi}{16}(2\text{x}+1 )^{2}.2=\frac{27\pi}{8}(2\text{x} + 1)^{2}$
$\therefore $ Radius of the balloon $=\frac{3}{4}(2\text{x}+1)$
$\therefore$ Volume of the balloon $=\frac{4}{3}\pi\Big(\frac{3}{4}\pi2\text{x}+1\Big)=\frac{9\pi}{16}(2\text{x}+1)^3$ cu. units
$\therefore$ Rate of change of volume $\text{w.r.t.x}\ = \frac{\text{dV}}{\text{dx}}= \frac{9\pi}{16}.3(2\text{x}+1)^2 .\frac {\text{d}}{\text{dx}}(2\text{x}+\text{x})$
$=\frac{ 27\pi}{16}(2\text{x}+1 )^{2}.2=\frac{27\pi}{8}(2\text{x} + 1)^{2}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Prove that $y = \frac{{4\sin \theta }}{{\left( {2 + \cos \theta } \right)}} - \theta $ is an increasing function of $\theta$ in $\left[ {0,\frac{\pi }{2}} \right]$
→In the differential equation show that it is homogeneous and solve it $y^2+\left( x ^2- xy \right) \frac{d y}{d x}=0$.
→A coin is tossed 5 times. What is the probability that head appears an even number of times?
For the principal values, evaluate the following:
$\sin^{-1}\Big(-\frac{\sqrt3}{2}\Big)+\cos^{-1}\Big(\frac{\sqrt3}{2}\Big)$
→$\sin^{-1}\Big(-\frac{\sqrt3}{2}\Big)+\cos^{-1}\Big(\frac{\sqrt3}{2}\Big)$
Find the direction cosines of the line passing through two points $(-2, 4, -5)$ and $(1, 2, 3)$.
→Show that $f(x) = x^9 + 4x^7 + 11$ is an increasing function for all $\text{x}\in\text{R}.$.
→If with reference to the right handed system of mutually perpendicular unit vectors $\hat{i}, \hat{j}$ and $\hat{k}, \vec{\alpha}=3 \hat{i}-\hat{j.}$ $\vec{\beta}=2 \hat{i}+\hat{j}-3 \hat{k.}$ then express $\vec{\beta}$ in the form $\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2$, where $\vec{\beta}_1$ is $\|$ to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$.
→Using vector method, prove that the point is collinear:
A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1)
A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1)
In a legislative assembly election, a political group hired a public relations firm to promote its candidate in three ways: telephone, house calls, and letters. The cost per contact (in paise) is given in matrix A as
The number of contacts of each type made in two cities X and Y is given by
Find the total amount spent by the group in the two cities X and Y.
The number of contacts of each type made in two cities X and Y is given by
Find the total amount spent by the group in the two cities X and Y.
Find the equation of the plane passing through the following point:
(-5, 0, -6), (-3, 10, -9) and (-2, 6, -6)
(-5, 0, -6), (-3, 10, -9) and (-2, 6, -6)