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An element with mass 910 grams decays by
16.6% per minute. How much of the element is remaining after 20 minutes, to the nearest 1oth of a gram?
Answer:

An element with mass $910$ grams decays by $16.6 \%$ per minute. How much of the element is remaining after $20$ minutes, to the nearest $1$ oth of a gram? $\newline$ Answer:

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Full solution

Q. An element with mass

910

grams decays by

16.6 \%

per minute. How much of the element is remaining after

20

minutes, to the nearest

1

oth of a gram?

\newline

Answer:

Understand Problem and Formula: Understand the problem and determine the formula to use. $\newline$ We know that the element decays by $16.6\%$ per minute. This means that each minute, the element retains $100\% - 16.6\% = 83.4\%$ of its mass from the previous minute. To find the remaining mass after $20$ minutes, we can use the formula for exponential decay: $\newline$ Remaining mass = Initial mass $\times (1 - \text{decay rate}) ^ \text{time}$ $\newline$ where the decay rate is $16.6\%$ or $0.166$ , and time is $20$ minutes.
Convert Decay Rate: Convert the decay rate to a decimal and calculate the remaining percentage each minute. $\newline$ Decay rate as a decimal = $\frac{16.6\%}{100} = 0.166$ $\newline$ Remaining percentage each minute = $1 - 0.166 = 0.834$
Apply Exponential Decay: Apply the exponential decay formula to calculate the remaining mass after $20$ minutes. $\newline$ Remaining mass = $910$ grams $\times (0.834)^{20}$
Calculate Remaining Mass: Calculate the remaining mass using the values from the previous step. $\newline$ Remaining mass = $910 \times (0.834)^{20}$
Perform Calculation: Perform the calculation. $\newline$ Remaining mass $\approx 910 \times (0.834)^{20} \approx 910 \times 0.049 \approx 44.59$ grams
Round Remaining Mass: Round the remaining mass to the nearest $10$ th of a gram. $\newline$ Rounded remaining mass = $44.6$ grams

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