Question

question
an element with a mass of 370 grams decays by 17.6% per minute. to the nearest tenth of a minute, how long will it be until there are 90 grams of the element remaining?
answer attempt 2 out of 2

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 rate of decay (in decimal), and \( t \) is the time. Here, \( P = 370 \), \( A = 90 \), and \( r = 0.176 \). So we have the equation \( 90 = 370(1 - 0.176)^t \).

Step2: Simplify the equation

First, calculate \( 1 - 0.176 = 0.824 \). So the equation becomes \( 90 = 370(0.824)^t \). Then, divide both sides by 370: \( \frac{90}{370} = (0.824)^t \). Simplify \( \frac{90}{370} \approx 0.2432 \). So we have \( 0.2432 = (0.824)^t \).

Step3: Take the natural logarithm of both sides

Take \( \ln \) of both sides: \( \ln(0.2432) = \ln((0.824)^t) \). Using the logarithm power rule \( \ln(a^b) = b\ln(a) \), we get \( \ln(0.2432) = t\ln(0.824) \).

Step4: Solve for \( t \)

Divide both sides by \( \ln(0.824) \): \( t = \frac{\ln(0.2432)}{\ln(0.824)} \). Calculate the logarithms: \( \ln(0.2432) \approx -1.310 \), \( \ln(0.824) \approx -0.194 \). Then \( t = \frac{-1.310}{-0.194} \approx 6.75 \). Rounding to the nearest tenth, \( t \approx 6.8 \).

Answer:

\( 6.8 \)