A spherical raindrop evaporates at a rate pro | vedclass

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

Question

(C) Let $V$ be the volume and $S$ be the surface area of the spherical raindrop. We are given that the rate of evaporation is proportional to the surf

Vedclass · Accepted Answer

$r=3-t$

Vedclass · Answer

(C) Let $V$ be the volume and $S$ be the surface area of the spherical raindrop. We are given that the rate of evaporation is proportional to the surface area,so $\frac{dV}{dt} = -kS$,where $k > 0$ is a constant.Since $V = \frac{4}{3}\pi r^3$,we have $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$.The surface area $S = 4\pi r^2$.Substituting these into the rate equation: $4\pi r^2 \frac{dr}{dt} = -k(4\pi r^2)$.This simplifies to $\frac{dr}{dt} = -k$.Integrating with respect to $t$,we get $r = -kt + c$.At $t = 0$,$r = 3$,so $3 = -k(0) + c \Rightarrow c = 3$.Thus,$r = -kt + 3$.At $t = 1$,$r = 2$,so $2 = -k(1) + 3 \Rightarrow k = 1$.Substituting $k=1$ and $c=3$ into the expression for $r$,we get $r = -t + 3$ or $r = 3 - t$.

$A$ spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally is $3 \text{ mm}$ and $1 \text{ hour}$ later has been reduced to $2 \text{ mm}$,then the expression of radius $r$ of the raindrop at any time $t$ is (where $0 \leq t < 3$):

Similar Questions

$A$ spherical rain drop evaporates at a rate proportional to its surface area. If initially its radius is $3 \ mm$ and after $1 \ second$ it is reduced to $2 \ mm$,then at any time $t$ its radius is (where $0 \leq t < 3$)

The family of curves whose $x$ and $y$ intercepts of a tangent at any point are respectively double the $x$ and $y$ coordinates of that point is

The differential equation $y \frac{dy}{dx} + x = c$ represents

An ice ball melts at a rate proportional to the amount of ice present at that instant. Half of the initial quantity of ice melts in $15 \text{ minutes}$. Let $x_0$ be the initial quantity of ice. If after $30 \text{ minutes}$ the amount of ice left is $k x_0$,then the value of $k$ is:

The solution of the equation $\frac{x^2 d^2y}{dx^2} = \ln x$,given that at $x = 1$,$y = 0$ and $\frac{dy}{dx} = -1$,is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module