Question

section 2.10: applied optimization (homework)
score: 90/150 answered: 9/15
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question 10
0/10 pts 4 99 details
the concentration of a drug in the bloodstream (c(t)) at any time (t), in hours, is described by the equation
c(t)=\frac{100t}{t^{2}+16}
where (t = 0) corresponds to the time at which the drug was swallowed. determine how long it takes the drug to reach its maximum concentration.
it will take (square) hours until it reaches its maximum concentration.

Explanation:

Step1: Differentiate $C(t)$ using quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = 100t$, $u^\prime=100$, $v=t^{2}+16$, $v^\prime = 2t$. So, $C^\prime(t)=\frac{100(t^{2}+16)-100t\times(2t)}{(t^{2}+16)^{2}}=\frac{100t^{2}+1600 - 200t^{2}}{(t^{2}+16)^{2}}=\frac{1600 - 100t^{2}}{(t^{2}+16)^{2}}$.

Step2: Set $C^\prime(t)=0$ to find critical points

Set $\frac{1600 - 100t^{2}}{(t^{2}+16)^{2}}=0$. Since the denominator $(t^{2}+16)^{2}>0$ for all real $t$, we only need to solve $1600 - 100t^{2}=0$. Rearranging gives $100t^{2}=1600$, then $t^{2}=16$, so $t = 4$ or $t=-4$. But $t\geq0$ (because $t$ represents time since the drug was swallowed), so we discard $t=-4$.

Answer:

$4$