Question

the concentration of anesthesia in a patients bloodstream can be modeled by c(t) = 30t/t^2 + 4 where c is the concentration as a percent and t is the time in hours. find the y-intercept of c(t) interpret this value in the context of this problem. Find lim C(t) x > infinity and interpret your answer in the context of this problem

Answer 1

`\textsf{a)}\quad C(0) = 0`

`\textsf{b)}\quad \displaystyle\lim_(t \to \infty) \left((30t)/(t^2 + 4)\right)= 0`

Explanation:

The concentration of anesthesia in a patients bloodstream can be modeled by:

`C(t) = (30t)/(t^2 + 4)`

where C is the concentration (as a percent) and t is the time in hours.

Part (a)

The y-intercept is the point at which the curve crosses the y-axis, so when x = 0. Therefore, to find the y-intercept of C(t), substitute t = 0 into the function:

`C(0) = (30(0))/((0)^2 + 4)`

`C(0) = (0)/(4)`

`C(0) = 0`

In the context of this problem, it means that at t = 0 hours (which is the initial time), the concentration of anesthesia in the patient's bloodstream is 0%. This is expected because anesthesia is not present in the bloodstream before it is administered.

Part (b)

To find the limit as t approaches infinity, we need to evaluate:

`\displaystyle\lim_(t \to \infty) \left((30t)/(t^2 + 4)\right)`

Let f(t) = 30t

Let g(t) = t² + 4

Given t → ∞, evaluate at t = ∞:

f(∞) = ∞

g(∞) = ∞

As the limit of the ratio f(t)/g(t) as t approaches ∞ is an indeterminate form of ∞/∞, and the numerator and denominator are both differentiable, we can use L'Hôpital's rule to evaluate the limit.

`\boxed{\begin{array}{l}\underline{\textsf{L'H$\hat{\sf o}$pital's Rule}}\\\\\displaystyle\textsf{If}\; \lim_(x \to a)f(x)=\lim_(x \to a)g(x)=0,\;\textsf{or}\\\\\displaystyle\lim_(x \to a)f(x)=\pm \infty\;\;\textsf{and}\;\lim_(x \to a)g(x)=\pm \infty\\\\\textsf{then provided that}\;\displaystyle\lim_(x \to a)(f(x))/(g(x))\;\textsf{exists},\\\\\displaystyle\lim_(x \to a)(f(x))/(g(x))=\lim_(x \to a)(f'(x))/(g'(x))\\\\\end{array}}`

By L'Hôpital's rule:

`\displaystyle \begin{aligned}\lim_(t \to \infty)(30t)/(t^2+4)&=\lim_(t \to \infty) (30)/(2t)\\\\&=\lim_(t \to \infty)(15)/(t)\\\\&=15\lim_(t \to \infty)(1)/(t)\\\\&=15 \cdot (1)/(\infty)\\\\&=0\end{aligned}`

In the context of this problem, this means that as time (t) goes to infinity, the concentration of anesthesia in the patient's bloodstream approaches 0%. In other words, the anesthesia will become less and less concentrated in the bloodstream over time until it disappears.