A spherical balloon is being inflated at the | vedclass

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

Question

(A) Let $r$ be the radius of the spherical balloon. The volume $V$ of the sphere is given by $V = \frac{4}{3}\pi r^3$.Differentiating with respect to

Vedclass · Accepted Answer

$10$

Vedclass · Answer

(A) Let $r$ be the radius of the spherical balloon. The volume $V$ of the sphere is given by $V = \frac{4}{3}\pi r^3$.Differentiating with respect to time $t$,we get $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$.Given $\frac{dV}{dt} = 35 \, cm^3/min$,we have $35 = 4\pi r^2 \frac{dr}{dt}$,which implies $\frac{dr}{dt} = \frac{35}{4\pi r^2} \quad (1)$.The surface area $S$ of the sphere is $S = 4\pi r^2$.Differentiating with respect to $t$,we get $\frac{dS}{dt} = 8\pi r \frac{dr}{dt}$.Substituting the value of $\frac{dr}{dt}$ from $(1)$ into the expression for $\frac{dS}{dt}$:$\frac{dS}{dt} = 8\pi r \left( \frac{35}{4\pi r^2} \right) = \frac{70}{r}$.Given the diameter is $14 \, cm$,the radius $r = 7 \, cm$.Substituting $r = 7$ into the expression for $\frac{dS}{dt}$:$\frac{dS}{dt} = \frac{70}{7} = 10 \, cm^2/min$.

$A$ spherical balloon is being inflated at the rate of $35 \, cm^3/min$. The rate of increase in the surface area (in $cm^2/min$) of the balloon when its diameter is $14 \, cm$,is

Similar Questions

$A$ ladder $5 \ m$ long rests against a vertical wall. If its top slides downwards at the rate of $10 \ cm/s$,then the angle between the ladder and the floor is decreasing at the rate of how many $rad/s$ when its lower end is $4 \ m$ away from the wall?

$A$ particle is moving in a straight line according to the equation $s = 45t + 11t^2 - t^3$. The time at which it will come to rest is ......... $sec$.

The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^{3} - 0.003x^{2} + 15x + 4000$. Find the marginal cost when $17$ units are produced.

The sides of an equilateral triangle are increasing at the rate of $2 \, cm/sec$. The rate at which the area increases,when the side is $10 \, cm$ is:

$A$ spherical balloon is filled with $4500 \pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 \pi$ cubic meters per minute,then the rate (in meters per minute) at which the radius of the balloon decreases $49$ minutes after the leakage has begun,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module