An element with mass $650$ grams decays by $6 \%$ per minute. How much of the element is remaining after $8$ minutes, to the nearest $1$ oth of a gram? $\newline$ Answer:

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Exponential growth and decay: word problems

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Q. An element with mass

650

grams decays by

6 \%

per minute. How much of the element is remaining after

8

minutes, to the nearest

1

oth of a gram?

\newline

Answer:

Convert to Decimal: We need to calculate the remaining mass of the element after it decays by $6\%$ per minute for $8$ minutes. The decay process can be modeled by an exponential decay function. $\newline$ To find the remaining mass after each minute, we can use the formula: $\newline$ Remaining mass $=$ Initial mass $\times (1 - \text{decay rate})^{\text{time}}$ $\newline$ where the decay rate is $6\%$ or $0.06$ as a decimal, and time is the number of minutes.
Apply Decay Rate: First, convert the decay rate percentage to a decimal by dividing by $100$ . Decay rate = $6\% = \frac{6}{100} = 0.06$
Calculate Remaining Mass: Now, apply the decay rate to the initial mass for $8$ minutes using the formula. $\newline$ Remaining mass after $8$ minutes = $650 \times (1 - 0.06)^8$
Find Decay Factor: Calculate the remaining mass. $\newline$ Remaining mass after $8$ minutes = $650 \times (0.94)^8$
Multiply Initial Mass: Use a calculator to find $(0.94)^8$ . $(0.94)^8 \approx 0.629856$
Perform Multiplication: Multiply the initial mass by the decay factor to find the remaining mass. $\newline$ Remaining mass after $8$ minutes $\approx 650 \times 0.629856$
Round Remaining Mass: Perform the multiplication to find the remaining mass. $\newline$ Remaining mass after $8$ minutes $\approx 409.4064$ grams
Round Remaining Mass: Perform the multiplication to find the remaining mass. $\newline$ Remaining mass after $8$ minutes $\approx 409.4064$ gramsRound the remaining mass to the nearest tenth of a gram. $\newline$ Remaining mass after $8$ minutes $\approx 409.4$ grams