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An element with mass 570 grams decays by
15.2% per minute. How much of the element is remaining after 16 minutes, to the nearest 1oth of a gram?
Answer:

An element with mass $570$ grams decays by $15.2 \%$ per minute. How much of the element is remaining after $16$ minutes, to the nearest $1$ oth of a gram? $\newline$ Answer:

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Full solution

Q. An element with mass

570

grams decays by

15.2 \%

per minute. How much of the element is remaining after

16

minutes, to the nearest

1

oth of a gram?

\newline

Answer:

Understand the problem: Understand the problem and determine the formula to use. $\newline$ We know that the element decays by $15.2\%$ per minute. This means that each minute, the element retains $100\% - 15.2\% = 84.8\%$ of its mass from the previous minute. To find the remaining mass after $16$ minutes, we will apply the decay percentage repeatedly for $16$ times.
Convert to decimal: Convert the decay percentage to a decimal to use in calculations. $\newline$ $15.2\%$ as a decimal is $\frac{15.2}{100} = 0.152$ . Therefore, the remaining percentage as a decimal is $1 - 0.152 = 0.848$ .
Calculate remaining mass: Calculate the remaining mass after each minute. $\newline$ We will use the formula for exponential decay, which is: $\newline$ Remaining mass = Initial mass $\times (1 - \text{decay rate})^{\text{number of time periods}}$ $\newline$ In this case, the initial mass is $570$ grams, the decay rate is $0.152$ , and the number of time periods is $16$ minutes.
Perform calculation: Perform the calculation using the formula. $\newline$ Remaining mass after $16$ minutes = $570 \times (0.848)^{16}$
Calculate exact value: Calculate the exact value using a calculator. $\newline$ Remaining mass after $16$ minutes $\approx 570 \times (0.848)^{16} \approx 570 \times 0.049 \approx 27.93$ grams
Round the result: Round the result to the nearest tenth of a gram. $\newline$ The remaining mass after $16$ minutes, rounded to the nearest tenth of a gram, is approximately $27.9$ grams.

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