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How much of a radioactive kind of chromium will be left after $3$ months if you start with $64$ grams and the half-life is $1$ month? $\newline$ $\newline$ ____ grams $\newline$

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Exponential growth and decay: word problems

Full solution

Q. How much of a radioactive kind of chromium will be left after

3

months if you start with

64

grams and the half-life is

1

month?

\newline

\newline

____ grams

\newline

Identify initial values: Identify the initial amount of the substance, the total time that has passed, and the half-life period of the substance. $\newline$ Initial amount $a$ = $64$ grams $\newline$ Total time $t$ = $3$ months $\newline$ Half-life period $h$ = $1$ month
Use half-life formula: Use the half-life formula to calculate the remaining amount of the substance after a given time. The formula is $y = a \cdot \left(\frac{1}{2}\right)^{\frac{t}{h}}$ , where $y$ is the remaining amount, $a$ is the initial amount, $t$ is the total time, and $h$ is the half-life period.
Substitute known values: Substitute the known values into the formula. $y = 64 \times \left(\frac{1}{2}\right)^{\frac{3}{1}}$
Simplify the exponent: Simplify the exponent in the formula. $\newline$ $(\frac{3}{1})$ simplifies to $3$ , so the formula becomes $y = 64 \times (\frac{1}{2})^3$ .
Calculate remaining quantity: Calculate the remaining quantity of the radioactive substance. $\newline$ $y = 64 \times \left(\frac{1}{2}\right)^3$ $\newline$ $y = 64 \times \frac{1}{8}$ $\newline$ $y = 8$

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