The radius of an air bubble is increasing at | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The air bubble is in the shape of a sphere.The volume $V$ of a sphere with radius $r$ is given by $V = \frac{4}{3} \pi r^3$.Differentiating both s

Vedclass · Accepted Answer

$2 \pi 	ext{ cm}^3/	ext{s}$

Vedclass · Answer

(A) The air bubble is in the shape of a sphere.The volume $V$ of a sphere with radius $r$ is given by $V = \frac{4}{3} \pi r^3$.Differentiating both sides with respect to time $t$,we get:$\frac{dV}{dt} = \frac{4}{3} \pi \cdot 3r^2 \cdot \frac{dr}{dt} = 4 \pi r^2 \frac{dr}{dt}$.Given that $\frac{dr}{dt} = \frac{1}{2} 	ext{ cm/s}$ and $r = 1 	ext{ cm}$.Substituting these values into the equation:$\frac{dV}{dt} = 4 \pi (1)^2 \left( \frac{1}{2} ight) = 2 \pi 	ext{ cm}^3/	ext{s}$.Thus,the volume of the bubble is increasing at the rate of $2 \pi 	ext{ cm}^3/	ext{s}$.

The radius of an air bubble is increasing at the rate of $\frac{1}{2} \text{ cm/s}$. At what rate is the volume of the bubble increasing when the radius is $1 \text{ cm}$?

Similar Questions

The distance $s$ in meters covered by a particle in $t$ seconds is given by $s = 2 + 27t - t^3$. The particle will stop after covering a distance of:

Two particles $A$ and $B$ move from rest along a straight line with constant accelerations $f$ and $f'$ respectively. If $A$ takes $m$ seconds more than that of $B$ and describes $n$ units more than that of $B$ in acquiring the same velocity, then:

The rate of change of the volume of a sphere with respect to its surface area,when its radius is $2 \text{ cm}$,is

The volume of a cube is increasing at the rate of $8 \, cm^{3}/s$. How fast is the surface area increasing when the length of an edge is $12 \, cm$?

Water is being poured into a cylindrical vessel with a radius of $2 \text{ cm}$ at a rate of $8 \text{ cm}^3/\text{s}$. The rate at which the height of the water is increasing is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module