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An element with a mass of 350 grams decays by
6.6% per minute. To the nearest minute, how long will it be until there are 230 grams of the element remaining?
Answer:

An element with a mass of $350$ grams decays by $6.6 \%$ per minute. To the nearest minute, how long will it be until there are $230$ 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

350

grams decays by

6.6 \%

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

230

grams of the element remaining?

\newline

Answer:

Determine Decay Formula: Determine the decay formula. $\newline$ The decay of an element can be described by an exponential decay formula, which is $P(t) = P_0 \cdot e^{-kt}$ , where $P_0$ is the initial amount, $P(t)$ is the amount at time $t$ , $k$ is the decay constant, and $e$ is the base of the natural logarithm.
Convert to Decay Constant: Convert the percentage decay rate to a decay constant. $\newline$ The decay rate is given as $6.6\%$ per minute, which means that $k = 0.066$ per minute because $6.6\% = \frac{6.6}{100} = 0.066$ .
Set Up Equation: Set up the equation with the given values. $\newline$ We have $P_0 = 350$ grams, $P(t) = 230$ grams, and $k = 0.066$ . The equation becomes $230 = 350 \times e^{(-0.066t)}$ .
Solve for t: Solve for t. $\newline$ To isolate $t$ , we first divide both sides by $350$ , getting $\frac{230}{350} = e^{-0.066t}$ . Then we take the natural logarithm of both sides to get $\ln(\frac{230}{350}) = \ln(e^{-0.066t}) = -0.066t$ .
Calculate Time: Calculate the time $t$ . We continue the calculation: $t = \ln(\frac{230}{350}) / -0.066$ . First, calculate the natural logarithm of the ratio $\frac{230}{350}$ , which is approximately $\ln(0.6571)$ . Then divide by $-0.066$ to find $t$ .
Perform Calculations: Perform the calculations. $\newline$ Using a calculator, we find $\ln(0.6571) \approx -0.4196$ . Then we divide $-0.4196$ by $-0.066$ to get $t \approx 6.3561$ minutes.
Round to Nearest Minute: Round the time to the nearest minute. $\newline$ Since the question asks for the time to the nearest minute, we round $6.3561$ to $6$ minutes.

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