Question

The rate at which ice-ball melts is proportional to the amount of ice in it. If half of it melts in $20$ minutes, the amount of ice left after $40$ minutes compared to it original amount is

• A. $\left(\frac{1}{8}\right)th$
• B. $\left(\frac{1}{16}\right)th$
• C. $\left(\frac{1}{4}\right)th$
• D. $\left(\frac{1}{32}\right)th$

Answer 1

The correct option is C $\left(\frac{1}{4}\right)th$

Rate at which ice melts$\propto$amount of ice

$\frac{dx}{dt}\propto x$

if x is amount of ice and t, time

$\frac{dx}{x}=kdt$

where, k is any constant

${\log }_{e}x=kt$

$x=e^{kt}$

Now, when $x=\frac{1}{2},t=20$ minutes

$\frac{1}{2}=e^{20k}$

$e^k={\left(\frac{1}{2}\right)}^{1/20}$

Now,

when $t=40$

$x=e^{kt}$

$x=e^{40t}$

$x={\left(e^k\right)}^{40}$

$x={\left(\frac{1}{2}\right)}^{40/20}$

$x={\left(\frac{1}{2}\right)}^2$

$x=\frac{1}{4}$

Therefore, amount of ice left is ${\left(\frac{1}{4}\right)}^{th}$ of original amount.