Consider the following scenario: A sample of radioactive material has a decay constant of 0.01 per day. If there are initially 100 grams of the material, what equation would you use to determine how much will remain after 50 days?

Question

1)y=50(0.99)^100
2)y=50(0.01)^100
3)y=100(0.99)^50
4)y=100(0.01)^50

Answer 1

To determine the remaining amount of a radioactive substance after a given period, you can use the exponential decay formula:

\[ N(t) = N_0 e^{-\lambda t} \]

Where:

• \( N(t) \) is the amount remaining after time \( t \),
• \( N_0 \) is the initial amount of the substance,
• \( \lambda \) is the decay constant,
• \( t \) is the time in the same units as the decay constant.

In your scenario:

• The initial amount \( N_0 = 100 \) grams,
• The decay constant \( \lambda = 0.01 \) per day,
• The time \( t = 50 \) days.

Alternatively, a simplified version of the equation can be expressed using the half-life or the property that the remaining amount can also be calculated as:

\[ N(t) = N_0 (1 - p)^t \]

where \( p = e^{-\lambda} \) is the proportion remaining after one time period (in this case, one day). In your case, since \( 1 - p \) is very close to \( 0.01 \) per day:

Using this simplified form, we want to calculate:

\[ y = 100(0.99)^{50} \]

Thus, the correct equation from the options provided, where you replace the base with \( 0.99 \) raised to the power of the number of days, looks like:

Answer: 3) \( y = 100(0.99)^{50} \)