Answer:
[tex]\textsf{(a)}\quad n(t)=8600e^{0.15082t}\quad \textsf{or}\quad n(t)=8600e^{t\ln\left(\frac{50}{43}\right)}[/tex]
(b) 11,628
(c) 278 minutes
Step-by-step explanation:
Part (a)
The general form of an exponential function is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{General form of an Exponential Function with base $e$}}\\\\f(t)=ae^{kt}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$f(t)$ is the value of the function at time $t$.}\\\phantom{ww}\bullet\;\textsf{$a$ is the initial value ($y$-intercept).} \\ \phantom{ww}\bullet\;\textsf{$e$ is Euler's number.}\\\phantom{ww}\bullet\;\textsf{$k$ is a constant.}\\ \phantom{ww}\bullet\;\textsf{$t$ is time.}\end{array}}[/tex]
Given that a culture starts with 8600 bacteria, then a = 8600.
If after 1 hour the count is 10000, then y = 10000 when t = 1.
Substitute these values into the general equation and solve for k:
[tex]8600e^{k\cdot 1}=10000\\\\\\8600e^k=10000\\\\\\e^k=\dfrac{50}{43}\\\\\\\ln(e^k)=\ln\left(\dfrac{50}{43}\right)\\\\\\k\ln(e)=\ln\left(\dfrac{50}{43}\right)\\\\\\k=\ln\left(\dfrac{50}{43}\right)\\\\\\k=0.15082\; \sf (5\;d.p.)[/tex]
Therefore, the function that models the number of bacteria after t hours is:
[tex]n(t)=8600e^{0.15082t}[/tex]
The exact form is:
[tex]n(t)=8600e^{t\ln\left(\frac{50}{43}\right)}[/tex]
[tex]\hrulefill[/tex]
Part (b)
To find the number of bacteria after 2 hours, substitute t = 2 into the function from part (a):
[tex]n(2)=8600e^{0.15082\cdot 2}\\\\\\n(2)=8600e^{0.30164}\\\\\\n(2)=11627.839773...\\\\\\n(2)=11628[/tex]
Note: If we substitute t = 2 into the exact function, we get 11627.906976 which still rounds to 11628.
[tex]\hrulefill[/tex]
Part (c)
To find the time when the bacteria doubles, set n(t) equal to 17200 and solve for t:
[tex]8600e^{0.15082t}=17200\\\\\\e^{0.15082t}=2\\\\\\\ln (e^{0.15082t})=\ln(2)\\\\\\0.15082t\ln(e)=\ln(2)\\\\\\0.15082t=\ln(2)\\\\\\t=\dfrac{\ln(2)}{0.15082}\\\\\\t=4.59585718445...\; \sf hours[/tex]
To convert the time to minutes, multiply the number of hours by 60:
[tex]t=275.7514310...\textsf{minutes}\\\\\\t=278\; \textsf{minutes}[/tex]
Note: If we use the exact function, we get t = 4.5957691288 hours, which converts to 275.74614772 minutes, and therefore still rounds to 278 minutes.
