Final answer:
To estimate the doubling time for a bacterial population modeled by N = 250e^{kt}, we solve for k using the given data (N=280 when t=10) and then use the doubling condition (N=500) to solve for the time t required for the population to double in size from the initial 250.
Step-by-step explanation:
The student wants to estimate the time required for a bacterial population to double in size based on the exponential growth model N = 250ekt. Given that N=280 when t=10, we first need to solve for k to understand the growth rate:
280 = 250e10k
Divide both sides by 250: e10k = 280/250
Take the natural logarithm of both sides: 10k = ln(280/250)
Solve for k: k = (ln(280/250))/10
To find the time it takes for the population to double (500 bacteria in this case), we set N = 2×250 and use the value of k we found:
500 = 250ekt
Divide both sides by 250: 2 = ekt
Take the natural logarithm of both sides: ln(2) = kt
Solve for t: t = ln(2)/k
By substituting the calculated value of k into t, we estimate the doubling time for the bacterial culture.