Answer:
415.63 minutes
Step-by-step explanation:
Growth can be represented by the equation [tex]A=A_0e^{rt}[/tex]. We can find the rate at which it grows by using t=25 minutes and [tex]\frac{A_{0}}{A} =2[/tex] or double the amount at that time. The first step we always take is to divide [tex]A_0[/tex] by A.
[tex][tex]\frac{A_{0}}{A}=e^{rt}\\2=e^{r(25)}[/tex]
[tex]2=e^{(25)r}[/tex]
To solve for r, we will take the natural log of both sides and use log rules to isolate r.
[tex]ln 2=ln e^{(25)r}\\ln 2=25r (ln e)\\\frac{ln2}{25} =r[/tex]
We know [tex]lne=1[/tex] so we were able to cancel it out and divide both sides by 25.
We solve with a calculator [tex]\frac{ln2}{25} =r\\0.0277=r[/tex]
We change 0.0277 into a percent by multiplying by 100 to get 2.77% as the rate.
The equation is [tex]A=A_0e^{0.0277t}[/tex] .
We repeat the step above substituting A=5,000,000, [tex]A_0[/tex]=50, and r=0.02777. Then solve for t.
[tex]5000000=50e^{0.0277t}\\\frac{5000000}{50} =e^{0.0277t}\\100000=e^{0.0277t}\\ln100000=lne^{0.0277t}\\ln100000=0.02777t(lne)\\\frac{ln100000}{0.0277} =t[/tex]
t=415.63 minutes
