To determine how much of the radioactive material will remain after 3 hours, you can use the exponential decay formula:
\[ y = y_0 e^{-\lambda t} \]
Where:
- \(y_0\) is the initial amount of the substance (500 grams),
- \(\lambda\) is the decay constant (0.04 per hour),
- \(t\) is the time in hours (3 hours),
- \(e\) is the base of the natural logarithm.
However, based on the options provided, we will also consider using a simplified version of the equation, which is common in decay problems:
\[ y = y_0 (1 - r)^t \]
Where:
- \(r\) is the decay fraction per time period. Since the decay constant \(\lambda = 0.04\), the decay fraction is \(r = e^{-\lambda} = e^{-0.04}\).
First, let's compute the decay factor:
1. Selecting the correct equation: In the context of the options given, we can approximate the decay per hour as follows:
- The decay factor for one hour is approximately \(1 - 0.04 = 0.96\).
This gives the equation as: \[ y = 500(0.96)^t \] So for \(t = 3\), the equation becomes: \[ y = 500(0.96)^3 \]
2. Calculating the remaining material after 3 hours: Now we calculate \(y\):
\[ y = 500(0.96)^3 \] Calculating \( (0.96)^3 \):
\[ (0.96)^3 \approx 0.941192 \]
Then: \[ y \approx 500 \times 0.941192 \approx 470.596 \]
Finally, rounding to the nearest tenth: \[ y \approx 470.6 \text{ grams} \]
So to summarize:
- The correct equation to use is \(y = 500(0.96)^3\).
- The amount of material remaining after 3 hours is approximately 470.6 grams.