Full solution

Q. An element with mass

650

grams decays by

29.5 \%

per minute. How much of the element is remaining after

20

minutes, to the nearest

10

th of a gram?

\newline

Answer:

Identify: Identify the initial amount, decay rate per minute, and total time. $\newline$ Initial amount $a$ = $650$ grams $\newline$ Decay rate per minute $r$ = $29.5\%$ $\newline$ Total time $t$ = $20$ minutes $\newline$ We need to calculate the remaining amount of the element after $20$ minutes.
Convert Rate: Convert the decay rate from a percentage to a decimal. $\newline$ To convert a percentage to a decimal, divide by $100$ . $\newline$ $r = \frac{29.5\%}{100} = 0.295$
Calculate Percentage: Calculate the remaining percentage after each minute. $\newline$ Since the element decays by $29.5\%$ each minute, it retains $100\% - 29.5\% = 70.5\%$ of its mass each minute. $\newline$ Remaining percentage per minute = $100\% - 29.5\% = 70.5\%$ $\newline$ Convert this to a decimal: $70.5\% / 100 = 0.705$
Use Exponential Decay: Use the exponential decay formula to calculate the remaining amount after $20$ minutes. $\newline$ The formula for exponential decay is: $\newline$ remaining amount = initial amount $\times$ $(1 - \text{decay rate})^{\text{time}}$ $\newline$ Substitute the values into the formula: $\newline$ remaining amount = $650 \times (0.705)^{20}$
Calculate Remaining Amount: Calculate the remaining amount using the values from Step $4$ . $\newline$ $\text{remaining amount} = 650 \times (0.705)^{20}$ $\newline$ Use a calculator to find $(0.705)^{20}$ . $\newline$ $(0.705)^{20} \approx 0.016258$ $\newline$ Now, multiply this by the initial amount: $\newline$ $\text{remaining amount} \approx 650 \times 0.016258$ $\newline$ $\text{remaining amount} \approx 10.5677$
Round to Nearest: Round the remaining amount to the nearest $10$ th of a gram. $\newline$ The remaining amount rounded to the nearest $10$ th of a gram is approximately $10.6$ grams.