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The half-life of a radioactive kind of praseodymium is $17$ minutes. If you start with $56$ grams of it, how much will be left after $34$ minutes? $\newline$ ____ grams

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Exponential growth and decay: word problems

Full solution

Q. The half-life of a radioactive kind of praseodymium is

17

minutes. If you start with

56

grams of it, how much will be left after

34

minutes?

\newline

____ grams

Given values: We have: $\newline$ Initial amount: $56$ grams $\newline$ Total time: $34$ minutes $\newline$ Half-life period: $17$ minutes $\newline$ Identify the values of $a$ , $t$ and $h$ . $\newline$ Initial amount $(a) = 56$ grams $\newline$ Total time $(t) = 34$ minutes $\newline$ Half-life period $(h) = 17$ minutes
Substitute values in formula: We found: $\newline$ $a = 56$ $\newline$ $t = 34$ $\newline$ $h = 17$ $\newline$ Identify the formula of half-life after substituting the values. $\newline$ Substitute $a = 56$ , $t = 34$ , and $h = 17$ in $y = a \cdot \left(\frac{1}{2}\right)^{\frac{t}{h}}$ . $\newline$ $y = 56 \cdot \left(\frac{1}{2}\right)^{\frac{34}{17}}$
Simplify exponent: Let's simplify the exponent in $y = 56 \times (\frac{1}{2})^{\frac{34}{17}}$ . Simplify $\frac{34}{17}$ . $\frac{34}{17} = 2$
Calculate remaining quantity: We have: $\newline$ $y = 56 \times (\frac{1}{2})^2$ $\newline$ Calculate the remaining quantity of the radioactive substance. $\newline$ $y = 56 \times (\frac{1}{2})^2$ $\newline$ $= 56 \times \frac{1}{4}$ $\newline$ $= 14$ $\newline$ Remaining quantity of praseodymium after $34$ minutes: $14$ grams

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