To solve for the remaining amount of a radioactive material after a given time, we can use the exponential decay formula:
\[ y = y_0 e^{-\lambda t} \]
where:
- \( y \) is the remaining amount of the substance,
- \( y_0 \) is the initial amount,
- \( \lambda \) is the decay constant,
- \( t \) is the time that has passed.
In this case:
- \( y_0 = 500 \) grams,
- \( \lambda = 0.04 \) per hour,
- \( t = 3 \) hours.
However, we can also express the remaining amount using the alternative form of the decay model, which is:
\[ y = y_0 (1 - e^{-\lambda})^t \]
Given that \( e^{-\lambda} \) can be approximated using \( 1 - \lambda \) for small values of \( \lambda \), we can make our calculations easier.
But, the simpler method is derived from the decay constant:
- The fraction remaining after \( t \) hours is \( e^{-\lambda t} \).
Calculating this step by step:
- Calculate \( e^{-\lambda t} = e^{-0.04 \cdot 3} = e^{-0.12} \).
- We can approximate \( e^{-0.12} \approx 0.888 \) using a calculator or constants table.
Thus remaining amount will be:
\[ y = 500 \cdot e^{-0.12} \approx 500 \cdot 0.888 \approx 444 \] which can be computed directly as follows as per the decay to factor:
Using \( e^{-\lambda} = 1 - 0.04 \) approximates very close to type (0.96), then:
\[ y = 500 \cdot (0.96)^3 \]
Calculating \( (0.96)^3 \):
\[ (0.96)^3 \approx 0.884736. \] So, plugging it back gives:
\[ y \approx 500 \times 0.884736 \approx 442.368 \text{ grams} \]
Now looking back at the responses you provided, we can check which matches:
The correct expression that you'll look for in response choices is:
y = 500(0.96)^3
This means after 3 hours approximately 442.37 grams would remain.