Question

The rate of decay of a certain substance is directly proportional to the amount present at that instant. Initially, there are 27 grams of the substance and 3 hours later it is found that 8 grams are left, then the amount left after one more hour is

• A. \( \frac{19}{3} \) grams
• B. \( \frac{20}{3} \) grams
• C. \( \frac{17}{3} \) grams
• D. \( \frac{16}{3} \) grams

Hint:

In exponential decay problems, use the formula \( A(t) = A_0 e^{-kt} \) and solve for \( k \) using known values. Then substitute back to find the remaining amount.

Answer 1

The Correct Option is D

Step 1: Using the exponential decay formula.
The exponential decay formula is given by: \[ A(t) = A_0 e^{-kt} \] where \( A_0 \) is the initial amount, \( k \) is the decay constant, and \( t \) is the time.

Step 2: Applying the given information.
Initially, there are 27 grams of the substance, so \( A_0 = 27 \). After 3 hours, 8 grams are left. Using the formula: \[ 8 = 27 e^{-3k} \] Solving for \( k \), we get: \[ e^{-3k} = \frac{8}{27}, \quad -3k = \ln \left( \frac{8}{27} \right), \quad k = \frac{-\ln \left( \frac{8}{27} \right)}{3} \]
Step 3: Finding the amount left after 4 hours.
Now, we substitute \( k \) into the equation for \( A(t) \) to find the amount left after 4 hours: \[ A(4) = 27 e^{-4k} \] After calculating this, we find that \( A(4) = \frac{16}{3} \) grams.

Step 4: Conclusion.
Thus, the amount left after 4 hours is \( \frac{16}{3} \) grams, which makes option (D) the correct answer.