Question

1. what is the initial amount of a substance if it decays to 250 grams in 10 years, with a decay rate of 4% per year?

a. 376 grams
b. 752 grams
c. 250 grams
d. 169 grams

Explanation:

Step1: Recall the exponential decay formula

The formula for exponential decay is \( A = P(1 - r)^t \), where \( A \) is the final amount, \( P \) is the initial amount, \( r \) is the decay rate (in decimal), and \( t \) is the time in years.
We know \( A = 250 \), \( r = 0.04 \) (since 4% = 0.04), and \( t = 10 \). We need to solve for \( P \).
Rearranging the formula to solve for \( P \), we get \( P=\frac{A}{(1 - r)^t} \).

Step2: Substitute the known values into the formula

Substitute \( A = 250 \), \( r = 0.04 \), and \( t = 10 \) into the formula:
\( P=\frac{250}{(1 - 0.04)^{10}}=\frac{250}{(0.96)^{10}} \)

Step3: Calculate \( (0.96)^{10} \)

Using a calculator, \( (0.96)^{10}\approx0.664832959 \)

Step4: Calculate \( P \)

\( P=\frac{250}{0.664832959}\approx376 \) (rounded to the nearest whole number)

Answer:

a. 376 grams