(C) The radioactive decay follows the formula $N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$,where $N_0$ is the initial mass,$t$ is the tim

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(C) The radioactive decay follows the formula $N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$,where $N_0$ is the initial mass,$t$ is the time elapsed,and $T_{1/2}$ is the half-life period.Given: $N_0 = 1000 \text{ mg}$,$t = 50 \text{ days}$,and $T_{1/2} = 10 \text{ days}$.The number of half-lives $n = \frac{t}{T_{1/2}} = \frac{50}{10} = 5$.The remaining mass $N(t) = 1000 \times \left(\frac{1}{2}\right)^5$.$N(t) = 1000 \times \frac{1}{32} = \frac{1000}{32} \text{ mg}$.Dividing both numerator and denominator by $8$,we get $N(t) = \frac{125}{4} \text{ mg}$.

Class 12 Mathematics Differential Equations Application of differential equations

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$A$ radioactive sample has a half-life period of $10 \text{ days}$. If a sample originally has a mass of $1000 \text{ mg}$,then the mass remaining after $50 \text{ days}$ is:

MHT CET 2020 All MHT CET

A
$\frac{225}{8} \text{ mg}$
B
$\frac{125}{8} \text{ mg}$
C
$\frac{125}{4} \text{ mg}$
D
$\frac{225}{4} \text{ mg}$

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$A$ radioactive sample has a half-life period of $10 \text{ days}$. If a sample originally has a mass of $1000 \text{ mg}$,then the mass remaining after $50 \text{ days}$ is:

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The rate of change of volume of a spherical balloon at any instant is directly proportional to its surface area. If initially its radius is $3 \ cm$,and after $2 \ minutes$ its radius becomes $9 \ cm$,then what is the radius of the balloon after $4 \ minutes$ (in $cm$)?

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Verify that the function $y=c_{1} e^{a x} \cos b x+c_{2} e^{a x} \sin b x,$ where $c_{1}, c_{2}$ are arbitrary constants,is a solution of the differential equation $\frac{d^{2} y}{d x^{2}}-2 a \frac{d y}{d x}+\left(a^{2}+b^{2}\right) y=0$.

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