An element with a mass of $910$ grams decays by $27.4 \%$ per minute. To the nearest minute, how long will it be until there are $10$ grams of the element remaining? $\newline$ Answer:

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Exponential growth and decay: word problems

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Q. An element with a mass of

910

grams decays by

27.4 \%

per minute. To the nearest minute, how long will it be until there are

10

grams of the element remaining?

\newline

Answer:

Identify amounts and rate: Identify the initial amount, final amount, and the decay rate per minute. $\newline$ Initial amount $a$ = $910$ grams $\newline$ Final amount $y$ = $10$ grams $\newline$ Decay rate per minute = $27.4\%$
Convert rate to decimal: Convert the decay rate from a percentage to a decimal for calculation purposes. $\newline$ Decay rate per minute = $27.4\% = \frac{27.4}{100} = 0.274$
Use exponential decay formula: Use the exponential decay formula $y = a \times (1 - r)^t$ , where $y$ is the final amount, $a$ is the initial amount, $r$ is the decay rate, and $t$ is the time in minutes. $\newline$ We need to solve for $t$ when $y = 10$ grams, $a = 910$ grams, and $r = 0.274$ . $\newline$ $10 = 910 \times (1 - 0.274)^t$
Isolate exponential part: Divide both sides of the equation by $910$ to isolate the exponential part of the equation. $\newline$ $\frac{10}{910} = (1 - 0.274)^t$
Simplify left side: Simplify the left side of the equation. $\newline$ $\frac{10}{910} \approx 0.01098901$ $\newline$ $0.01098901 = (1 - 0.274)^t$
Take natural logarithm: Take the natural logarithm ( $\ln$ ) of both sides to solve for $t$ . $\ln(0.01098901) = \ln((1 - 0.274)^t)$
Bring down exponent: Use the property of logarithms that $\ln(a^b) = b \times \ln(a)$ to bring down the exponent $t$ . $\ln(0.01098901) = t \times \ln(1 - 0.274)$
Calculate logarithms: Calculate $\ln(1 - 0.274)$ and $\ln(0.01098901)$ using a calculator. $\newline$ $\ln(1 - 0.274) \approx \ln(0.726) \approx -0.319$ $\newline$ $\ln(0.01098901) \approx -4.512$
Solve for t: Divide both sides of the equation by $\ln(1 - 0.274)$ to solve for t. $t = \frac{\ln(0.01098901)}{\ln(1 - 0.274)}$ $t \approx \frac{-4.512}{-0.319}$
Round to nearest minute: Calculate the value of $t$ using the values from the previous step. $\newline$ $t \approx -4.512 / -0.319$ $\newline$ $t \approx 14.144$
Round to nearest minute: Calculate the value of $t$ using the values from the previous step. $t \approx \frac{-4.512}{-0.319}$ $t \approx 14.144$ Since we cannot have a fraction of a minute in this context, we round $t$ to the nearest whole minute. $t \approx 14$ minutes