Question

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 $Kx_0$, then K =

• A. $\frac{1}{2}$
• B. $\frac{1}{8}$
• C. $\frac{1}{4}$
• D. $\frac{1}{3}$

Hint:

This problem perfectly models first-order radioactive half-life decay patterns! If the half-life of the process is 20 minutes, then 1 half-life passes at 20 minutes (leaving $\frac{1}{2}$), and exactly 2 half-lives pass at 40 minutes. The remaining fraction is simply $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$ instantly!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem states that the rate of decrease of ice volume over time is directly proportional to the amount of ice remaining at that instant. We are given the time taken for half the ice to melt (20 minutes), and we need to determine the fractional constant $K$ representing the amount left after 40 minutes.

Step 2: Key Formula or Approach:
Let $x$ be the quantity of ice remaining at time $t$. The word description translates into a first-order separable differential equation: $$ \frac{dx}{dt} = -kx $$ where $k$ is a positive constant of proportionality, and the negative sign indicates that the amount of ice is shrinking over time.

Step 3: Detailed Explanation:
Separating variables and integrating both sides of our differential equation: $$ \int \frac{dx}{x} = -k \int dt \implies \ln x = -kt + C $$ Exponentiating both sides allows us to write the solution in standard exponential decay form: $$ x(t) = x_0 e^{-kt} $$ where $x_0$ is the initial quantity at $t = 0$. Let's apply the first boundary condition: half of the ice melts in 20 minutes, meaning half remains at $t = 20$: $$ x(20) = \frac{x_0}{2} \implies \frac{x_0}{2} = x_0 e^{-20k} \implies e^{-20k} = \frac{1}{2} $$ We need to find the remaining amount after $t = 40$ minutes, given as $Kx_0$: $$ Kx_0 = x_0 e^{-40k} \implies K = e^{-40k} $$ Using the exponent rule $(e^{-20k})^2 = e^{-40k}$, we substitute our known fractional value into the equation: $$ K = \left(e^{-20k}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} $$

Step 4: Final Answer:
The value of the constant $K$ is $\frac{1}{4}$, which corresponds to option (C).