Question

Consider the following scenario:
A sample of radioactive material has a decay constant of 0.02 per day. If there are initially 20 grams of the material, what formula would you use to determine how much will remain after 10 days?
A. y=10(0.02)
30

B. y=20(0.02)
30

C. y=10(0.98)
20

D. y=20(0.98)
10

Answer 1

To determine the amount of radioactive material remaining after a certain time, you can use the exponential decay formula:

\[
y = y_0 e^{-\lambda t}
\]

where:
- \( y \) is the amount remaining after time \( t \),
- \( y_0 \) is the initial amount,
- \( \lambda \) is the decay constant, and
- \( t \) is the time in days.

In this case, you have:
- The initial amount \( y_0 = 20 \) grams,
- The decay constant \( \lambda = 0.02 \) per day, and
- The time \( t = 10 \) days.

Using the decay formula, we can rewrite it using exponential decay in terms of the decay factor, which is \( (1 - \lambda) \):

\[
y = y_0 (1 - \lambda)^t = y_0 (0.98)^t
\]

Substituting in the values:

\[
y = 20 (0.98)^{10}
\]

Thus, the correct formula to determine how much material will remain after 10 days is:

**D. \( y = 20(0.98)^{10} \)**.

The options provided might contain a typographical error since none of them match the standard form above. However, from the context, **D** is the closest correct form to represent the remaining amount after the decay over 10 days.