Question
Consider the following scenario: A sample of radioactive material has a decay constant of 0.04 per hour. If there are initially 500 grams of the material, how much will remain after 3 hours?(1 point) Responses y=500(0.96)3 y is equal to 500 times 0 point 9 6 cubed y=3(0.96)500 y is equal to 3 times 0 point 9 6 to the 500th power y=3(0.04)500 y is equal to 3 times 0 point 0 4 to the 500th power y=500(0.04)3
Answers
GPT-4o mini
To determine the remaining amount of radioactive material after a certain time, you 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 in hours,
- \(e\) is Euler's number.
Given:
- \(y_0 = 500\) grams,
- \(\lambda = 0.04\) per hour,
- \(t = 3\) hours.
First, you can calculate \(e^{-\lambda t}\):
\[
y = 500 e^{-0.04 \times 3} = 500 e^{-0.12}
\]
Next, calculate \(e^{-0.12}\):
\[
e^{-0.12} \approx 0.8869
\]
Now, substitute that value back into the equation:
\[
y \approx 500 \times 0.8869 \approx 443.45 \text{ grams}
\]
So the exact form using the decay factor \(0.96\) (since \(e^{-0.04} \approx 0.96\)) gives:
\[
y = 500(0.96^3)
\]
Calculating that:
\[
0.96^3 \approx 0.884736
\]
Thus,
\[
y \approx 500 \times 0.884736 \approx 442.368 \text{ grams}
\]
However, from the provided options, the equivalent expression that represents the solution is:
**\( y = 500(0.96^3) \)**
This matches with the first response you provided: **\(y = 500(0.96)^3\)**.
\[
y = y_0 e^{-\lambda t}
\]
where:
- \(y\) is the remaining amount of the substance,
- \(y_0\) is the initial amount,
- \(\lambda\) is the decay constant,
- \(t\) is the time in hours,
- \(e\) is Euler's number.
Given:
- \(y_0 = 500\) grams,
- \(\lambda = 0.04\) per hour,
- \(t = 3\) hours.
First, you can calculate \(e^{-\lambda t}\):
\[
y = 500 e^{-0.04 \times 3} = 500 e^{-0.12}
\]
Next, calculate \(e^{-0.12}\):
\[
e^{-0.12} \approx 0.8869
\]
Now, substitute that value back into the equation:
\[
y \approx 500 \times 0.8869 \approx 443.45 \text{ grams}
\]
So the exact form using the decay factor \(0.96\) (since \(e^{-0.04} \approx 0.96\)) gives:
\[
y = 500(0.96^3)
\]
Calculating that:
\[
0.96^3 \approx 0.884736
\]
Thus,
\[
y \approx 500 \times 0.884736 \approx 442.368 \text{ grams}
\]
However, from the provided options, the equivalent expression that represents the solution is:
**\( y = 500(0.96^3) \)**
This matches with the first response you provided: **\(y = 500(0.96)^3\)**.