Enhanced Homework
Part 6 of 8 where $A_{0}$ is the initial amount of the medication. Assume that an initial amount of $150 \mathrm{cc}$ is injected. Complete parts (a) through (d).
\[
A(t)=\frac{A_{0}}{t^{2}+1}
\]
a) Find $A(0), A(1), A(2), A(7)$, and $A(10)$.
\[
A(0)=150.0 \quad \mathrm{cc}
\]
(Type an integer or decimal rounded to the nearest ten-thousandth as needed.)
\[
A(1)=75.0 \quad c c
\]
(Type an integer or decimal rounded to the nearest ten-thousandth as needed.)
\[
A(2)=30.0 \quad \mathrm{cc}
\]
(Type an integer or decimal rounded to the nearest ten-thousandth as needed.)
\[
A(7)=3.0 \quad \mathrm{cc}
\]
(Type an integer or decimal rounded to the nearest ten-thousandth as needed.)
\[
A(10)=1.4851 \quad c c
\]
(Type an integer or decimal rounded to the nearest ten-thousandth as needed.)
b) Find the maximum amount of medication in the bloodstream over the interval $[0, \infty)$.
The maximum amount of medication is at $\mathrm{t}=$

Step 1 ：The function given is \(A(t)=\frac{A_{0}}{t^{2}+1}\), where \(A_{0}\) is the initial amount of the medication. In this case, \(A_{0}=150 \mathrm{cc}\).

Step 2 ：To find the values of the function at specific points, we substitute the values of \(t\) into the function. For \(t=0, 1, 2, 7, 10\), the values are \(150.0 \mathrm{cc}, 75.0 \mathrm{cc}, 30.0 \mathrm{cc}, 3.0 \mathrm{cc}, 1.4851 \mathrm{cc}\) respectively.

Step 3 ：To find the maximum value of the function over the interval \([0, \infty)\), we find the derivative of the function and set it equal to zero to find the critical points. The maximum value of the function will occur at one of these critical points or at the endpoints of the interval.

Step 4 ：The derivative of the function is \(-300t/(t^{2} + 1)^{2}\). Setting this equal to zero gives one critical point at \(t=0\).

Step 5 ：The maximum value of the function over the interval \([0, \infty)\) is \(150 \mathrm{cc}\), which occurs at \(t=0\). This makes sense because the amount of medication in the bloodstream is highest immediately after the injection and then decreases over time.

Step 6 ：Final Answer: The values of the function \(A(t)\) at the points \(t=0, 1, 2, 7, 10\) are \(150.0 \mathrm{cc}, 75.0 \mathrm{cc}, 30.0 \mathrm{cc}, 3.0 \mathrm{cc}, 1.4851 \mathrm{cc}\) respectively. The maximum amount of medication in the bloodstream over the interval \([0, \infty)\) is \(\boxed{150.0 \mathrm{cc}}\), which occurs at \(t=0\).