Select the correct answer.
A scientist is studying the decay of a certain substance after being exposed to two different treatments. Under treatment A, the substance continuously decays at a rate of $4 \%$ daily. Under treatment B, another sample of the same substance continuously decays at a rate of $6.2 \%$ daily.
A second scientist comes to record the amount remaining each day and only knows that there was initially less than 300 grams of the substance undergoing treatment A and at most 400 grams of the substance undergoing treatment B.
Which system of inequalities can be used to determine $t$, the number of days after which the remaining amount of each sample, $y$, in grams, is the same?
$\left\{\begin{array}{l}
y \leq 300 e^{-0.04 t} \\
y<400 e^{-0.062 t}
\end{array}\right.$
$\left\{\begin{array}{l}
y \leq 300 e^{0.04 t} \\
y<400 e^{0.062 t}
\end{array}\right.$
$\left\{\begin{array}{l}
y<300 e^{-0.04 t} \\
y \leq 400 e^{-0.062 t}
\end{array}\right.$
$\left\{\begin{array}{l}
y<300 e^{0.04 t} \\
y \leq 400 e^{0.062 t}
\end{array}\right.$
Answer (2)
The problem involves modeling the decay of a substance under two different treatments using exponential decay formulas.
Treatment A has an initial amount less than 300 grams and a decay rate of 4%, leading to the inequality y < 300 e − 0.04 t .
Treatment B has an initial amount at most 400 grams and a decay rate of 6.2%, leading to the inequality y ≤ 400 e − 0.062 t .
The system of inequalities that models the situation is { y < 300 e − 0.04 t y ≤ 400 e − 0.062 t .
Explanation
Understanding the Problem We are given that a substance decays continuously under two different treatments. Treatment A has a decay rate of 4% daily, and the initial amount is less than 300 grams. Treatment B has a decay rate of 6.2% daily, and the initial amount is at most 400 grams. We need to find the system of inequalities that models this situation, where y represents the remaining amount of the substance and t represents the number of days.
Modeling Treatment A The general formula for continuous decay is given by y = A e − k t , where A is the initial amount, k is the decay rate, and t is the time. For treatment A, since the initial amount is less than 300 grams, we have A < 300 . The decay rate is 4%, so k = 0.04 . Therefore, the inequality for treatment A is y < 300 e − 0.04 t .
Modeling Treatment B For treatment B, the initial amount is at most 400 grams, so A ≤ 400 . The decay rate is 6.2%, so k = 0.062 . Therefore, the inequality for treatment B is y ≤ 400 e − 0.062 t .
Combining the Inequalities Combining the inequalities for both treatments, we get the system of inequalities: { y < 300 e − 0.04 t y ≤ 400 e − 0.062 t
Final Answer Comparing this system of inequalities with the given options, we find that the correct answer is: { y < 300 e − 0.04 t y ≤ 400 e − 0.062 t
Examples
Understanding exponential decay is crucial in various real-world scenarios, such as determining the shelf life of medications or estimating the remaining amount of radioactive material over time. For instance, if a hospital stores a radioactive isotope with an initial amount of 200 grams and a decay rate of 5% per day, we can use the formula y = 200 e − 0.05 t to calculate the amount remaining after t days. This helps ensure safe handling and disposal of the material, highlighting the practical importance of exponential decay models.
The problem involves modeling the decay of substances under different treatments using exponential decay. The system of inequalities for Treatment A and Treatment B is represented as: { y < 300 e − 0.04 t , y ≤ 400 e − 0.062 t } . This allows for determining the time t where both substances have the same remaining amount.
;
