009A Sample Midterm 1, Problem 2 Detailed Solution

Suppose the size of a population at time  $t$  is given by

$N(t)={\frac {1000t}{5+t}},~t\geq 0.$

(a) Determine the size of the population as  $t\rightarrow \infty .$  We call this the limiting population size.

(b) Show that at time  $t=5,$  the size of the population is half its limiting size.

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Background Information:
Recall that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\rightarrow \infty} \frac{a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0}{b_nx^n+b_{n-1}x^{n-1}+\cdots +b_0}=\frac{a_n}{b_n}}
provided  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n\ne 0}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_n\ne 0.}

Solution:

(a)

Step 1:
We have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{t\rightarrow \infty} N(t)=\lim_{t\rightarrow \infty} \frac{1000t}{5+t}.}

Step 2:
Using the Background Information, we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\lim_{t\rightarrow \infty} N(t)} & = & \displaystyle{\frac{1000}{1}}\\ &&\\ & = & \displaystyle{1000.} \end{array}}

(b)
We have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{N(5)} & = & \displaystyle{\frac{1000(5)}{5+5}}\\ &&\\ & = & \displaystyle{\frac{1000(5)}{10}}\\ &&\\ & = & \displaystyle{100(5)}\\ &&\\ & = & \displaystyle{500}\\ &&\\ & = & \displaystyle{\frac{1000}{2}.} \end{array}}

Final Answer:
(a)     Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1000}
(b)     Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N(5)=500}

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