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An element with a mass of 870 grams decays by
18.3% per minute. To the nearest minute, how long will it be until there are 60 grams of the element remaining?
Answer:

An element with a mass of $870$ grams decays by $18.3 \%$ per minute. To the nearest minute, how long will it be until there are $60$ grams of the element remaining? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q. An element with a mass of

870

grams decays by

18.3 \%

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

60

grams of the element remaining?

\newline

Answer:

Determine decay rate per minute: Determine the decay rate per minute. $\newline$ The element decays by $18.3\%$ per minute, which means that each minute, the element retains $100\% - 18.3\% = 81.7\%$ of its mass.
Convert percentage to decimal: Convert the percentage to a decimal to use in calculations. $81.7\%$ as a decimal is $0.817$ .
Set up exponential decay formula: Set up the exponential decay formula. $\newline$ The formula for exponential decay is $P(t) = P_0 \times (1 - r)^t$ , where $P(t)$ is the final amount, $P_0$ is the initial amount, $r$ is the decay rate per period, and $t$ is the number of periods. In this case, we need to modify the formula to account for the decay rate as a remaining percentage: $P(t) = P_0 \times r^t$ .
Substitute values into formula: Substitute the known values into the decay formula. $\newline$ We want to find the time $t$ when $P(t) = 60$ grams, $P_0 = 870$ grams, and $r = 0.817$ . So we have $60 = 870 \times 0.817^t$ .
Solve for t: Solve for t. $\newline$ To isolate $t$ , we first divide both sides by $870$ : $\frac{60}{870} = 0.817^t$ .
Calculate left side of equation: Calculate the left side of the equation. $\frac{60}{870}$ equals approximately $0.0689655$ .
Take natural logarithm: Take the natural logarithm of both sides to solve for $t$ . $\ln(0.0689655) = t \times \ln(0.817)$ .
Calculate natural logarithm: Calculate the natural logarithm of both sides. $\newline$ $\ln(0.0689655) \approx -2.67415$ and $\ln(0.817) \approx -0.20397$ .
Divide by $\ln(0.817)$ : Divide by $\ln(0.817)$ to solve for $t$ . $\newline$ $t = \frac{-2.67415}{-0.20397}.$
Calculate value of t: Calculate the value of t. $t \approx \frac{-2.67415}{-0.20397} \approx 13.109$ .
Round $t$ to nearest minute: Round $t$ to the nearest minute. $\newline$ Since we cannot have a fraction of a minute in this context, we round $t$ to the nearest whole number, which is $13$ minutes.

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