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Question: Answered & Verified by Expert
An ice ball melts at the rate which is proportional to the amount of ice at that instant. Half the quantity of ice melts in 20 minutes. $x_0$ is the initial quantity of ice. If after 40 minutes the amount of ice left is $k x_0 x$, then $k$ is
MathematicsDifferential EquationsMHT CETMHT CET 2022 (10 Aug Shift 1)
Options:
  • A $\frac{1}{8}$
  • B $\frac{1}{2}$
  • C $\frac{1}{3}$
  • D $\frac{1}{4}$
Solution:
1818 Upvotes Verified Answer
The correct answer is: $\frac{1}{4}$
$\begin{aligned} & \frac{\mathrm{d} x}{\mathrm{~d} t}=-k x \\ & \Rightarrow \frac{\mathrm{d} x}{x}=-k d t \\ & \Rightarrow \log _e x=-k t+c \\ & \Rightarrow x=e^{-k t+c}=e^c \cdot e^{-k t} \\ & \text { at } t=0, x^{-k}=x_0 \\ & \Rightarrow e^c=x_0 \\ & \Rightarrow x=x_0 e^{-k t}\end{aligned}$

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