Question

a population of butterflies decreases exponentially at a rate of 18.3% per year. what is the equivalent monthly rate to the nearest hundredth of a percent? 1.27% 1.41% 1.53% 1.67%

Explanation:

Step1: Recall the formula for exponential decay

The formula for exponential decay is \( A = P(1 - r)^t \), where \( r \) is the annual rate (in decimal) and \( t \) is time in years. For monthly rate, let the monthly rate be \( i \) (in decimal), and since there are 12 months in a year, the formula becomes \( A = P(1 - i)^{12} \). We know the annual rate \( r = 0.183 \) (18.3% in decimal), so the equation for the decay factor is \( 1 - r=(1 - i)^{12} \).

Step2: Solve for the monthly rate \( i \)

First, calculate \( 1 - r \): \( 1 - 0.183 = 0.817 \). So we have \( 0.817=(1 - i)^{12} \). To solve for \( 1 - i \), take the 12th root of both sides: \( 1 - i = 0.817^{\frac{1}{12}} \). Calculate \( 0.817^{\frac{1}{12}} \). Using a calculator, \( 0.817^{\frac{1}{12}}\approx0.9847 \). Then, solve for \( i \): \( i = 1 - 0.9847 = 0.0153 \). Convert this to a percentage: \( 0.0153\times100 = 1.53\% \).

Answer:

1.53%