Choose the one alternative that best completes the statement or answers the question.
-A nuclear scientist has a sample of $100 \mathrm { mg }$ of a radioactive material which has a half-life in hours. She monitors the amount of radioactive material over a period of a day and obtains the following data. Use a graphing utility to fit an exponential function to the data. Predict the amount of material remaining at 40 hours.
$\begin{array}{l|lcccccc}
\text { Hours } & 0 & 5 & 10 & 15 & 20 & 25 & 30 \
\hline \mathrm{mg} & 100 & 68.3 & 45.2 & 31.3 & 21.5 & 14.6 & 9.8
\end{array}$

A) $y = 92 \mathrm { e } ^ { - 0.0686 x } , 5.9 \mathrm { mg }$
B) $y = 86 e ^ { - 0.071 x , 5.0 \mathrm { mg } }$
C) $y = 100 \mathrm { e } ^ { - 0.077 x } , 4.6 \mathrm { mg }$
D) $y = 100 \mathrm { e } ^ { - 0.077 x } , 6.7 \mathrm { mg }$

Question

Choose the one alternative that best completes the statement or answers the question.
-A nuclear scientist has a sample of $100 \mathrm { mg }$ of a radioactive material which has a half-life in hours. She monitors the amount of radioactive material over a period of a day and obtains the following data. Use a graphing utility to fit an exponential function to the data. Predict the amount of material remaining at 40 hours.
$\begin{array}{l|lcccccc}
\text { Hours } & 0 & 5 & 10 & 15 & 20 & 25 & 30 \
\hline \mathrm{mg} & 100 & 68.3 & 45.2 & 31.3 & 21.5 & 14.6 & 9.8
\end{array}$

A) $y = 92 \mathrm { e } ^ { - 0.0686 x } , 5.9 \mathrm { mg }$
B) $y = 86 e ^ { - 0.071 x , 5.0 \mathrm { mg } }$
C) $y = 100 \mathrm { e } ^ { - 0.077 x } , 4.6 \mathrm { mg }$
D) $y = 100 \mathrm { e } ^ { - 0.077 x } , 6.7 \mathrm { mg }$

Quizplus · Accepted Answer

The exponential decay formula for radioactive material is $y = y_0 e^{-kt}$, where $y_0$ is the initial amount, $k$ is the decay constant, and $t$ is time. The option that correctly represents the decay of the material with the initial amount of 100 mg and fits the data provided is $y = 100 e^{-0.077x}$. To predict the amount of material remaining at 40 hours, substituting $x = 40$ into the equation gives approximately 4.6 mg, which matches option C.

Choose the One Alternative That Best Completes the Statement or Answers