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You have $80$ grams of a radioactive kind of tellurium. How much will be left after $8$ months if its half-life is $2$ months? $\newline$ ____ grams

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Exponential growth and decay: word problems

Full solution

Q. You have

80

grams of a radioactive kind of tellurium. How much will be left after

8

months if its half-life is

2

months?

\newline

____ grams

Identify Values: Identify the values of the initial amount $a$ , total time $t$ , and half-life period $h$ . $\newline$ Initial amount $a$ = $80$ grams $\newline$ Total time $t$ = $8$ months $\newline$ Half-life period $h$ = $2$ months
Use Half-life Formula: Use the half-life formula to determine the remaining amount of the substance after a given time. $\newline$ The formula is $y = a \times (1/2)^{(t/h)}$ , where $y$ is the remaining amount after time $t$ , $a$ is the initial amount, and $h$ is the half-life period. $\newline$ Substitute $a = 80$ , $t = 8$ , and $h = 2$ into the formula to get $y = 80 \times (1/2)^{(8/2)}$ .
Simplify Exponent: Simplify the exponent in the formula. $\newline$ Simplify $8/2$ to get $4$ . $\newline$ The formula now is $y = 80 \times (1/2)^4$ .
Calculate Remaining Quantity: Calculate the remaining quantity of the radioactive substance. $\newline$ $y = 80 \times (\frac{1}{2})^4$ $\newline$ $= 80 \times \frac{1}{16}$ $\newline$ $= 5$ $\newline$ Remaining quantity of tellurium after $8$ months: $5$ grams

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