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

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Aicharya · Accepted Answer

$\begin{aligned} & \frac{\mathrm{d} x}{\mathrm{~d} t}=-k x \ & \Rightarrow \frac{\mathrm{d} x}{x}=-k d t \ & \Rightarrow \log _e x=-k t+c \ & \Rightarrow x=e^{-k t+c}=e^c \cdot e^{-k t} \ & 	ext { at } t=0, x^{-k}=x_0 \ & \Rightarrow e^c=x_0 \ & \Rightarrow x=x_0 e^{-k t}\end{aligned}$

An ice ball melts at the rate which is proportional to the amount of ice at that instant. Half the quantity…

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