Question

suppose that 5 mg of a drug is injected into the bloodstream. let f(t) be the amount present in the bloodstream after t hours. in drug in the bloodstream after $2\frac{1}{4}$ hours.
what is the meaning of f(2) = 3?
a. 2 hours after the drug was injected, the amount present in the bloodstream is 3 mg.
b. 2 hours after the drug was injected, the amount present in the bloodstream is rising at a rate of 3 mg per hour.
c. 3 hours after the drug was injected, the amount present in the bloodstream is rising at a rate of 2 mg per hour.
d. 3 hours after the drug was injected, the amount present in the bloodstream is 2 mg.
what is the meaning of f(2) = -0.3?
a. 2 hours after the drug was injected, the amount present in the bloodstream is falling at a rate of 0.3 mg per hour.
b. 2 hours after the drug was injected, the amount present in the bloodstream is rising at a rate of 0.3 mg per hour.
c. 0.3 hour after the drug was injected, the amount present in the bloodstream is 2 mg.
d. 2 hours after the drug was injected, the amount present in the bloodstream is -0.3 mg.
after $2\frac{1}{4}$ hours, the number of milligrams of the drug in the bloodstream will be □ mg.
(type an integer or a decimal.)

Explanation:

Step1: Analyze the meaning of \(f(t)\)

Given \(f(t)\) is the amount of drug in the bloodstream after \(t\) hours. When \(t = 2\), \(f(2)=3\) means at \(t = 2\) (2 hours after injection), the value of the function (amount of drug) is 3 mg.

Step2: Analyze the meaning of \(f^{\prime}(t)\)

The derivative \(f^{\prime}(t)\) represents the rate of change of the amount of drug in the bloodstream with respect to time. When \(t = 2\) and \(f^{\prime}(2)=- 0.3\), the negative sign indicates a decrease. So it means 2 hours after injection, the amount of drug is falling at a rate of \(0.3\) mg per hour.

Step3: Calculate \(f(2\frac{1}{4})\)

Assume the function \(f(t)\) has a form \(f(t)=mt + b\) (a simple linear - approximation for illustration purposes, if we consider the initial value \(f(0) = 5\) (5mg injected initially), and from \(f(2)=3\), we can find the slope \(m=\frac{f(2)-f(0)}{2}=\frac{3 - 5}{2}=-1\). Then \(f(t)=-t + 5\).
Now, \(t = 2\frac{1}{4}=\frac{9}{4}\). Substitute \(t=\frac{9}{4}\) into \(f(t)\):
\(f(\frac{9}{4})=-\frac{9}{4}+5=\frac{-9 + 20}{4}=\frac{11}{4}=2.75\)

Answer:

For the first question: A. 2 hours after the drug was injected, the amount present in the bloodstream is 3 mg.
For the second question: A. 2 hours after the drug was injected, the amount present in the bloodstream is falling at a rate of 0.3 mg per hour.
For the third question: \(2.75\)