Question:
The rate of change of volume of spherical balloon at any instant is directly proportional to its surface area. If initially its radius is 3 cm, after 2 minutes its radius becomes 9 cm, then radius of balloon after 4 minutes is
The rate of change of volume of spherical balloon at any instant is directly proportional to its surface area. If initially its radius is 3 cm, after 2 minutes its radius becomes 9 cm, then radius of balloon after 4 minutes is
Show Hint
If $dV/dt \propto S$ for a sphere, the radius increases at a constant rate ($dr/dt = \text{const}$).
Updated On: Apr 30, 2026
- 12 cm
- 14 cm
- 15 cm
- 18 cm
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Formulate Equation
$dV/dt = kS \implies 4\pi r^2 (dr/dt) = k(4\pi r^2)$.
$dr/dt = k$.
Step 2: Integrate
$r = kt + C$.
At $t=0, r=3 \implies C = 3$.
At $t=2, r=9 \implies 9 = 2k + 3 \implies 2k = 6 \implies k = 3$.
Step 3: Calculate final radius
At $t=4$: $r = (3)(4) + 3$.
$r = 12 + 3 = 15$ cm.
Step 4: Conclusion
The radius is 15 cm.
Final Answer:(C)
$dV/dt = kS \implies 4\pi r^2 (dr/dt) = k(4\pi r^2)$.
$dr/dt = k$.
Step 2: Integrate
$r = kt + C$.
At $t=0, r=3 \implies C = 3$.
At $t=2, r=9 \implies 9 = 2k + 3 \implies 2k = 6 \implies k = 3$.
Step 3: Calculate final radius
At $t=4$: $r = (3)(4) + 3$.
$r = 12 + 3 = 15$ cm.
Step 4: Conclusion
The radius is 15 cm.
Final Answer:(C)
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Vectors Questions
- If $\vec{a} , \vec{b} , \vec{c}$ are mutually perpendicular vectors having magnitudes 1, 2, 3 respectively, then $[\vec{a} + \vec{b} + \vec{c} \, \, \vec{b} - \vec{a} - \vec{c}] = ?$
- If and are two probability density functions,Which one of the following statements is true?
Three vectors a, b and c are given. Find the equation of a vector that lies in the plane of vector a and vector b and whose projection on vector c is 1/√3.
- If \( \sqrt{\frac{y}{x}} + 4\sqrt{\frac{x}{y}} = 4 \), then \( \frac{dy}{dx} \):
- Evaluate \( \sin \left( \tan^{-1}\frac{4}{5} + \tan^{-1}\frac{4}{3} + \tan^{-1}\frac{1}{9} - \tan^{-1}\frac{1}{7} \right) \):
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions