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A radioactive compound with mass 420 grams decays at a rate of
4% per hour. Which equation represents how many grams of the compound will remain after 5 hours?

C=420(0.6)^(5)

C=420(1+0.04)^(5)

C=420(1-0.04)(1-0.04)(1-0.04)

C=420(0.96)^(5)

A radioactive compound with mass $420$ grams decays at a rate of $4 \%$ per hour. Which equation represents how many grams of the compound will remain after $5$ hours? $\newline$ $C=420(0.6)^{5}$ $\newline$ $C=420(1+0.04)^{5}$ $\newline$ $C=420(1-0.04)(1-0.04)(1-0.04)$ $\newline$ $C=420(0.96)^{5}$

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Write exponential functions: word problems

Full solution

Q. A radioactive compound with mass

420

grams decays at a rate of

4 \%

per hour. Which equation represents how many grams of the compound will remain after

5

hours?

\newline

C=420(0.6)^{5}

\newline

C=420(1+0.04)^{5}

\newline

C=420(1-0.04)(1-0.04)(1-0.04)

\newline

C=420(0.96)^{5}

Identify initial amount and decay rate: Identify the initial amount of the compound and the decay rate. $\newline$ The initial amount of the compound $a$ is $420$ grams. $\newline$ The decay rate is $4\%$ per hour, which means that each hour, the compound retains $96\%$ of its mass from the previous hour.
Convert decay rate to decimal: Convert the decay rate to a decimal. $\newline$ To convert a percentage to a decimal, divide by $100$ . $\newline$ $4\%$ as a decimal is $0.04$ .
Determine decay factor: Determine the decay factor. $\newline$ Since the compound decays by $4\%$ , it retains $100\% - 4\% = 96\%$ of its mass each hour. $\newline$ As a decimal, this is $0.96$ . $\newline$ The decay factor $(b)$ is therefore $0.96$ .
Write exponential decay equation: Write the exponential decay equation. $\newline$ The general form of an exponential decay equation is $C = a(b)^t$ , where: $\newline$ $C$ is the final amount, $\newline$ $a$ is the initial amount, $\newline$ $b$ is the decay factor ( $1$ minus the decay rate), $\newline$ $t$ is the time in hours. $\newline$ Substitute the known values into the equation to get: $\newline$ $C = 420(0.96)^t$
Calculate remaining amount after $5$ hours: Calculate the amount of the compound remaining after $5$ hours. $\newline$ Substitute $5$ for $t$ in the equation from Step $4$ to get: $\newline$ $C = 420(0.96)^5$

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