An ice ball melts at a rate proportional to t | vedclass

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(C) Let $x(t)$ be the amount of ice at time $t$. The rate of melting is proportional to the amount of ice,so $\frac{dx}{dt} = -cx$,where $c > 0$. Inte

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$1/4$

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(C) Let $x(t)$ be the amount of ice at time $t$. The rate of melting is proportional to the amount of ice,so $\frac{dx}{dt} = -cx$,where $c > 0$. Integrating this,we get $x(t) = x_0 e^{-ct}$. Given that half the ice melts in $15 	ext{ minutes}$,at $t = 15$,$x(15) = \frac{1}{2} x_0$. So,$\frac{1}{2} x_0 = x_0 e^{-15c}$,which implies $e^{-15c} = \frac{1}{2}$. We need to find the amount of ice left after $30 	ext{ minutes}$,which is $x(30) = x_0 e^{-30c}$. $x(30) = x_0 (e^{-15c})^2 = x_0 \left(\frac{1}{2}ight)^2 = \frac{1}{4} x_0$. Comparing this with $k x_0$,we get $k = \frac{1}{4}$.

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:

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