To determine how much of a radioactive material remains after a certain period, you can use the formula for exponential decay:
\[ y = y_0 \cdot e^{-\lambda t} \]
Where:
- \( y \) is the remaining amount of the material,
- \( y_0 \) is the initial amount of the material,
- \( \lambda \) is the decay constant,
- \( t \) is the time in days, and
- \( e \) is the base of the natural logarithm.
However, for this particular scenario, since you have a decay constant in a form that suggests you can use a simpler exponential decay formula based on the half-life concept, you can use the base (which represents the remaining fraction after one time unit).
Given a decay constant \( \lambda = 0.02 \) per day, the fraction that remains after \( t \) days can be expressed as:
\[ y = y_0 \cdot (e^{-\lambda t}) \]
Or if you prefer to use an approximation that remains within simpler calculations without using \( e \), you can also use:
\[ y = y_0 \cdot (1 - \text{decay rate})^t \]
Where the decay rate can be approximated as \( e^{-\lambda} \).
In your case:
- The initial amount \( y_0 \) is 20 grams,
- The decay constant \( \lambda = 0.02 \) per day,
- The time \( t = 10 \) days.
First, we need to establish the fraction of material that remains after one day:
\[ \text{decay fraction} = e^{-\lambda} = e^{-0.02} \] This is approximately:
\[ \text{decay fraction} \approx 0.9802 \]
For a simpler model, we can round it to 0.98. So the formula you would use to determine how much of the material will remain after 10 days is:
\[ y = 20(0.98)^{10} \]
Thus, the correct choice from your responses is:
y = 20(0.98)^10
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