An element with a mass of $220$ grams decays by $26.6 \%$ per minute. To the nearest tenth of a 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

220

grams decays by

26.6 \%

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

10

grams of the element remaining?

\newline

Answer:

Understand the decay process: Understand the decay process. $\newline$ The decay process can be described by the exponential decay formula: $N(t) = N_0 \times (1 - r)^t$ , where $N(t)$ is the remaining amount after time $t$ , $N_0$ is the initial amount, $r$ is the decay rate per time period, and $t$ is the time period.
Convert to decimal: Convert the percentage decay rate to a decimal. $\newline$ $26.6\%$ as a decimal is $0.266$ .
Set up equation: Set up the equation with the given values. $\newline$ We have $N_0 = 220$ grams, $N(t) = 10$ grams, and $r = 0.266$ . Plugging these into the decay formula gives us $10 = 220 \times (1 - 0.266)^t$ .
Simplify the equation: Simplify the equation. $\newline$ Divide both sides by $220$ to isolate the exponential expression: $\frac{10}{220} = (1 - 0.266)^t$ .
Calculate left side: Calculate the left side of the equation. $\frac{10}{220}$ simplifies to $\frac{1}{22}.$
Take natural logarithm: Take the natural logarithm of both sides to solve for $t$ . $\ln(\frac{1}{22}) = \ln((1 - 0.266)^t)$ .
Use logarithm property: Use the property of logarithms to bring down the exponent. $\newline$ $\ln(\frac{1}{22}) = t \times \ln(1 - 0.266)$ .
Calculate ln values: Calculate $\ln(1 - 0.266)$ and $\ln(\frac{1}{22})$ . $\newline$ $\ln(1 - 0.266) \approx \ln(0.734)$ and $\ln(\frac{1}{22}) \approx \ln(0.04545)$ .
Divide to solve: Divide to solve for $t$ . $t = \frac{\ln(0.04545)}{\ln(0.734)}$ .
Calculate t value: Calculate the value of $t$ . $t \approx \ln(0.04545) / \ln(0.734) \approx -3.09104 / -0.30952 \approx 9.987$ , which rounds to $10.0$ to the nearest tenth of a minute.