Questions: Once medicine X is injected, the amount of it in a person's bloodstream decays at a rate of 3.2% per hour. A patient was injected with a 250 mg dose of medicine X. Which formula below describes the amount of X remaining in the patient's bloodstream t hours after injection? - 250 * 1.032^(-t) - 250 * 0.968^t - -250 * 1.032^t - 250 * 0.032^t

Solution Steps

To determine the correct formula for the amount of medicine $X$ remaining in the bloodstream after $t$ hours, we need to consider the decay rate of $3.2\%$ per hour. The decay rate means that each hour, only $96.8\%$ of the medicine remains. Therefore, the correct formula should reflect this exponential decay.

The decay rate of the medicine is \(3.2\%\) per hour. This means that \(96.8\%\) of the medicine remains each hour. Therefore, the decay factor is \(0.968\).

The amount of medicine remaining after \(t\) hours can be modeled by the exponential decay formula: \[ A(t) = A_0 \cdot (0.968)^t \] where \(A_0\) is the initial dose and \(A(t)\) is the amount remaining after \(t\) hours.

Given: \[ A_0 = 250 \text{ mg} \] \[ t = 5 \text{ hours} \] Substitute these values into the formula: \[ A(5) = 250 \cdot (0.968)^5 \]

Using the given values: \[ A(5) \approx 250 \cdot 0.8499 \approx 212.4794 \text{ mg} \]

Final Answer