Question

score on last try: 2 of 5 pts. see details for more. at least one scored part is incorrect. jump to first changable incorrect part. > next question get a similar question you can retry this question below the count in a bacteria culture was 600 after 15 minutes and 1600 after 30 minutes. assuming the count grows exponentially ( f(t)=ce^{kt} ), find the initial size of the culture: 225 × (round to the nearest whole number.) find the growth rate: k = 0.065 √ (round to the nearest thousandth.) find the doubling period: t_d = 10.6 × minutes. (round to the nearest tenth.) find the population after 85 minutes: p = 58291 × (round to the nearest whole number.) the population will reach 13000 at t = 62.0 √ minutes. (round to the nearest tenth.)

Explanation:

Step1: Find the initial size $c$ and growth - rate $k$

We know that $f(t)=ce^{kt}$. When $t = 15$, $f(15)=ce^{15k}=600$; when $t = 30$, $f(30)=ce^{30k}=1600$.
Divide the second equation by the first equation: $\frac{ce^{30k}}{ce^{15k}}=\frac{1600}{600}$, which simplifies to $e^{15k}=\frac{8}{3}$.
Take the natural - logarithm of both sides: $15k=\ln(\frac{8}{3})$, so $k=\frac{\ln(\frac{8}{3})}{15}\approx0.065$.
Substitute $k$ into $ce^{15k}=600$, we have $c\times e^{15\times0.065}=600$, $c\times e^{0.975}=600$, $c=\frac{600}{e^{0.975}}\approx225$.

Step2: Find the doubling period $t_d$

We know that when the population doubles, $f(t + t_d)=2f(t)$. Using the formula $f(t)=ce^{kt}$, we have $ce^{k(t + t_d)}=2ce^{kt}$.
Cancel out $c$ and $e^{kt}$ on both sides, getting $e^{kt_d}=2$. Take the natural - logarithm of both sides: $kt_d=\ln(2)$, so $t_d=\frac{\ln(2)}{k}=\frac{\ln(2)}{0.065}\approx10.6$ minutes.

Step3: Find the population after 85 minutes

We use the formula $f(t)=ce^{kt}$, with $c = 225$, $k = 0.065$, and $t = 85$. Then $P=f(85)=225\times e^{0.065\times85}=225\times e^{5.525}\approx225\times251.37=56558.25\approx56558$.

Answer:

Initial size: 225
Growth rate: $k\approx0.065$
Doubling period: $t_d\approx10.6$ minutes
Population after 85 minutes: $P\approx56558$
Population reaches 13000 at $t\approx62.0$ minutes