To solve exponential growth problems, first determine the growth rate using given data points and the exponential growth formula. Then, use this rate to find the initial population size, doubling period, future population at a specific time, and the time it will take to reach a certain population size.
Using the exponential growth formula P(t)=P0ekt, we have two data points:
- P1=300 at t1=15
- P2=1000 at t2=35
By dividing the equations, we eliminate P0:
3001000=e20k⟹310=e20k
Taking the natural logarithm:
ln(310)=20k⟹k=20ln(310)≈0.060198640216296805
Now, substituting k back into the equation for P1:
P0=ekt1P1=e15k300≈121.6080139326331
Thus, the initial size of the culture is:
P0≈121.61
The doubling period T can be calculated using the formula:
T=kln(2)≈11.514332849868898
Thus, the doubling period is:
T≈11.51 minutes
To find the population after t3=75 minutes, we use:
P(75)=P0ek⋅75≈11111.111111111117
Thus, the population after 75 minutes is:
P(75)≈11111.11
- Initial size of the culture: 121.61
- Doubling period: 11.51 minutes
- Population after 75 minutes: 11111.11