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 rate of change of the surface area of a sphere of radius $r$ when the radius is increasing at the rate of $2 \text{ cm/sec}$ is proportional to

If the side of a cube is increased by $5 \%$,then the surface area of a cube is increased by (in $\%$)

$A$ balloon starting from rest is ascending from the ground with a uniform acceleration of $4 \ ft/sec^2$. At the end of $5 \ sec$, a stone is dropped from it. If $T$ is the time taken for the stone to reach the ground and $H$ is the height of the balloon when the stone reaches the ground, then:

$A$ particle moving from a fixed point on a straight line travels a distance $S$ metres in $t$ seconds. If $S = t^3 - t^2 - t + 3$, then the distance (in metres) travelled by the particle when it comes to rest is:

The rate of change of the diagonal $R$ of a square with respect to its area $A$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module