A balloon,which always remains spherical,has | vedclass

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

Question

(A) The volume $V$ of a sphere with radius $r$ is given by the formula: $V = \frac{4}{3} \pi r^3$. To find the rate at which the volume is increasing

Vedclass · Accepted Answer

$400 \pi 	ext{ cm}^3/	ext{cm}$

Vedclass · Answer

(A) The volume $V$ of a sphere with radius $r$ is given by the formula: $V = \frac{4}{3} \pi r^3$. To find the rate at which the volume is increasing with respect to the radius,we calculate the derivative $\frac{dV}{dr}$: $\frac{dV}{dr} = \frac{d}{dr} \left( \frac{4}{3} \pi r^3 ight) = \frac{4}{3} \pi (3r^2) = 4 \pi r^2$. We are asked to find this rate when the radius $r = 10 	ext{ cm}$. Substituting $r = 10$ into the derivative: $\frac{dV}{dr} = 4 \pi (10)^2 = 4 \pi (100) = 400 \pi$. Thus,the rate of change of volume with respect to the radius is $400 \pi 	ext{ cm}^3/	ext{cm}$.

$A$ balloon,which always remains spherical,has a variable radius. Find the rate at which its volume is increasing with respect to its radius when the radius is $10 \text{ cm}$.

Similar Questions

The displacement $s$ of a particle,in meters,at any time $t$ in seconds is expressed as $s = \frac{t^3}{3} - 6t$. Find the acceleration at the time when the velocity vanishes.

The radius of a circular plate is increasing at the rate of $0.01 \text{ cm/s}$ when the radius is $12 \text{ cm}$. Then,the rate at which the area increases,is

Displacement $s$ of a particle at time $t$ is expressed as $s=2 t^3-9 t$. Find the acceleration at the time when the velocity vanishes.

If the velocity of a moving particle is directly proportional to the square root of the distance it has covered,what is its acceleration?

$A$ ladder $5 \ m$ long is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of $2 \ m/sec$. How fast is its height on the wall decreasing when the foot of the ladder is $4 \ m$ away from the wall?

Vedclass Test Series

Exam Paper Generator

Online Exam Module