A radioactive compound with mass $180$ grams decays at a rate of $4 \%$ per hour. Which equation represents how many grams of the compound will remain after $8$ hours? $\newline$ $C=180(0.96)^{8}$ $\newline$ $C=180(1-0.04)(1-0.04)(1-0.04)$ $\newline$ $C=180(1+0.04)^{8}$ $\newline$ $C=180(1-0.4)^{8}$

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Exponential growth and decay: word problems

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Q. A radioactive compound with mass

180

grams decays at a rate of

4 \%

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

8

hours?

\newline

C=180(0.96)^{8}

\newline

C=180(1-0.04)(1-0.04)(1-0.04)

\newline

C=180(1+0.04)^{8}

\newline

C=180(1-0.4)^{8}

Understand the problem: Understand the problem. $\newline$ We need to find the equation that correctly represents the decay of a radioactive compound that starts with a mass of $180$ grams and decays at a rate of $4\%$ per hour over a period of $8$ hours.
Identify the correct decay formula: Identify the correct decay formula. $\newline$ The general formula for exponential decay is given by $C = C_0 \times (1 - r)^t$ , where $C_0$ is the initial amount, $r$ is the decay rate per unit time, and $t$ is the time.
Substitute values into formula: Substitute the given values into the decay formula. $\newline$ $C_0 = 180$ grams (initial amount), $r = 4\%$ per hour (decay rate), and $t = 8$ hours (time). $\newline$ Convert the percentage decay rate to a decimal: $4\% = 0.04$ .
Write equation with values: Write the equation using the values from Step $3$ . $C = 180 \times (1 - 0.04)^8$
Check for matching equation: Check the given options to see which one matches the equation from Step $4$ . $\newline$ The correct equation is $C = 180 \times (1 - 0.04)^8$ , which simplifies to $C = 180 \times (0.96)^8$ .
Verify other options: Verify that none of the other options match the correct equation. $\newline$ Option B: $C = 180 \times (1 - 0.04)(1 - 0.04)(1 - 0.04)$ is incorrect because it only accounts for three hours of decay, not eight. $\newline$ Option C: $C = 180 \times (1 + 0.04)^8$ is incorrect because it suggests growth, not decay. $\newline$ Option D: $C = 180 \times (1 - 0.4)^8$ is incorrect because the decay rate is $4\%$ , not $40\%$ .