(B) The rate of disintegration is given by $\frac{dM}{dt} = -kM$, where $k > 0$.Separating the variables, we get $\frac{dM}{M} = -k dt$.Integrating bo

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

(B) The rate of disintegration is given by $\frac{dM}{dt} = -kM$, where $k > 0$.Separating the variables, we get $\frac{dM}{M} = -k dt$.Integrating both sides, $\int \frac{dM}{M} = \int -k dt$, which gives $\ln M = -kt + C$.At $t = 0$, $M = M_0 = 1.5 \text{ g}$, so $C = \ln(1.5)$.Thus, $\ln M = -kt + \ln(1.5)$, or $\ln(\frac{M}{1.5}) = -kt$.We want to find $t$ when $M = 0.5 \text{ g}$.Substituting the values, $\ln(\frac{0.5}{1.5}) = -kt$.$\ln(\frac{1}{3}) = -kt$, which implies $-\ln 3 = -kt$.Therefore, $t = \frac{1}{k} \ln 3$ or $\frac{1}{k} \log_e 3$.

Class 12 Mathematics Differential Equations Application of differential equations

Difficult

The rate of disintegration of a radioactive element at any time $t$ is proportional to its mass $M$ at that time. If $k (k > 0)$ is the constant of proportionality, find the time required for an initial mass of $1.5 \text{ g}$ to disintegrate to a mass of $0.5 \text{ g}$.

MHT CET 2026 All MHT CET

A
$k \log 3$
B
$\frac{1}{k} \log 3$
C
$k \log 5$
D
$\frac{1}{k} \log 5$

Explore More

Browse All

Question Bank

All Subjects

Class 12 Questions

Class 12

Mathematics Questions

Chapter

Differential Equations

Subtopic

Application of differential equations

Previous Year

MHT CET 2026 Paper All MHT CET years →

Let the population of rabbits surviving at time $t$ be governed by the differential equation $\frac{dp(t)}{dt} = \frac{1}{2}p(t) - 200$. If $p(0) = 100$,then $p(t)$ equals:

DifficultJEE MAIN 2014

View Solution

$A$ spherical mothball has an initial radius of $3 \text{ cm}$. Due to evaporation, the radius of the ball reduces to $1 \text{ cm}$ in $4 \text{ months}$. In how many months would the mothball evaporate completely if the volume is lost at a rate proportional to the surface area?

DifficultMHT CET 2026

View Solution

The slope of the tangent at any point $(x, y)$ on the curve is equal to the product of the coordinates of that point. If the equation of the normal to the curve at the point $(\sqrt{2}, e)$ is $ax + by = 1$,then $\frac{b}{a} =$

EasyAP EAMCET 2023

View Solution

Let $f$ be a non-negative function defined in $[0, \pi / 2]$, $f^{\prime}$ exists and is continuous for all $x$, and $\int_0^x \sqrt{1-\left(f^{\prime}(t)\right)^2} dt = \int_0^x f(t) dt$ with $f(0) = 0$. Then

MediumWBJEE 2022

View Solution

The number of solutions of $\frac{dy}{dx} = \frac{y+1}{x-1}$,when $y(1) = 2$ is

DifficultKCET 2021

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

The rate of disintegration of a radioactive element at any time $t$ is proportional to its mass $M$ at that time. If $k (k > 0)$ is the constant of proportionality, find the time required for an initial mass of $1.5 \text{ g}$ to disintegrate to a mass of $0.5 \text{ g}$.

Similar Questions

Let the population of rabbits surviving at time $t$ be governed by the differential equation $\frac{dp(t)}{dt} = \frac{1}{2}p(t) - 200$. If $p(0) = 100$,then $p(t)$ equals:

$A$ spherical mothball has an initial radius of $3 \text{ cm}$. Due to evaporation, the radius of the ball reduces to $1 \text{ cm}$ in $4 \text{ months}$. In how many months would the mothball evaporate completely if the volume is lost at a rate proportional to the surface area?

The slope of the tangent at any point $(x, y)$ on the curve is equal to the product of the coordinates of that point. If the equation of the normal to the curve at the point $(\sqrt{2}, e)$ is $ax + by = 1$,then $\frac{b}{a} =$

Let $f$ be a non-negative function defined in $[0, \pi / 2]$, $f^{\prime}$ exists and is continuous for all $x$, and $\int_0^x \sqrt{1-\left(f^{\prime}(t)\right)^2} dt = \int_0^x f(t) dt$ with $f(0) = 0$. Then

The number of solutions of $\frac{dy}{dx} = \frac{y+1}{x-1}$,when $y(1) = 2$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module