MHT CET · Maths · Application of Derivatives
The rate of change of the volume of a sphere with respect to its surface area, when its radius is 2 cm, is _______ \(\mathrm{cm}^3 / \mathrm{cm}^2\).
- A 0.1
- B 0.5
- C 1
- D 2
Answer & Solution
Correct Answer
(C) 1
Step-by-step Solution
Detailed explanation
Volume of sphere \((V)=\frac{4}{3} \pi r^3\)
Surface area of sphere \((A)=4 \pi r^2\)
\(\begin{aligned}
& \therefore \quad \frac{d V}{d r}=4 \pi r^2 \text { and } \frac{d A}{d r}=8 \pi r \\
& \therefore \quad \frac{d V}{d A}=\frac{\frac{d V}{\frac{d r}{d A}}}{\frac{d r}{d r}}=\frac{4 \pi r^2}{8 \pi r}=\frac{r}{2} \\
& \therefore \quad\left(\frac{d V}{d A}\right)_{r=2}=\frac{2}{2}=1 \mathrm{~cm}^3 / \mathrm{cm}^2
\end{aligned}\)
Surface area of sphere \((A)=4 \pi r^2\)
\(\begin{aligned}
& \therefore \quad \frac{d V}{d r}=4 \pi r^2 \text { and } \frac{d A}{d r}=8 \pi r \\
& \therefore \quad \frac{d V}{d A}=\frac{\frac{d V}{\frac{d r}{d A}}}{\frac{d r}{d r}}=\frac{4 \pi r^2}{8 \pi r}=\frac{r}{2} \\
& \therefore \quad\left(\frac{d V}{d A}\right)_{r=2}=\frac{2}{2}=1 \mathrm{~cm}^3 / \mathrm{cm}^2
\end{aligned}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The angle between the linee \(\bar{r}=(i+\hat{j}-\hat{k})+\lambda(3 i+\hat{j})\) and the plane \(2+3 \hat{k})=8\)MHT CET 2020 Easy
- If an equation \(\mathrm{hxy}+\mathrm{g} x+\mathrm{f} y+\mathrm{c}=0\) represents a pair of lines, thenMHT CET 2024 Medium
- If \(e^x+e^y=e^{x+y}\), then \(\frac{d y}{d x}=\)MHT CET 2022 Medium
- If \(y=\left(\frac{x^{2}}{x+1}\right)^{x}\) and \(\frac{d y}{d x}=y\left[g(x)+\log \left(\frac{x^{2}}{x+1}\right)\right]\), then \(g(x)=\)MHT CET 2020 Easy
- Considering only the principal values of an inverse function, the set
\(\mathrm{A}=\left\{x \geq 0 / \tan ^{-1} x+\tan ^{-1} 6 x=\frac{\pi}{4}\right\}\)MHT CET 2023 Medium - The feasible region represented by the given constraints \(2 x+3 \mathrm{y} \geqslant 12,-x+\mathrm{y} \leqslant 3, x \leqslant 4, \mathrm{y} \geqslant 3\) is denoted by
MHT CET 2025 Medium
More PYQs from MHT CET
- A single turn current loop in the shape of a right angle triangle with side \(5 \mathrm{~cm}, 12 \mathrm{~cm}, 13 \mathrm{~cm}\) is carrying a current of \(2 \mathrm{~A}\). The loop is in a uniform magnetic field of magnitude \(0.75 \mathrm{~T}\) whose direction is parallel to the current in the \(13 \mathrm{~cm}\) side of the loop. The magnitude of the magnetic force on the \(5 \mathrm{~cm}\) side will be \(\frac{\mathrm{x}}{130} \mathrm{~N}\). The value of ' \(x\) ' isMHT CET 2023 Medium
- If \(\frac{1}{4}, \mathrm{a}, \mathrm{b}, \frac{1}{19}\) form a H.P. then the values of a and \(\mathrm{b}\) are respectivelyMHT CET 2020 Medium
- Consider a particle of mass m suspended by a string at the equator. Let R and M denote radius and mass of the earth. If is the angular velocity of rotation of the earth about its own axis, then the tension on the string will beMHT CET 2019 Hard
- What is the unit of viscosity?MHT CET 2020 Easy
- Which of the following reaction is used to prepare toluene from bromobenzene?MHT CET 2022 Easy
- The d.r.s. of the normal to the plane passing through the origin and the line of intersection of the planes \(x+2 y+3 z=4\) and \(4 x+3 y+2 z=1\) areMHT CET 2021 Medium