1) The number of bacteria N in a culture is given by N=250e^kt where t is the time in hours. If N = 280 when t=10 estimate the time required for the population to double in size.
280 = 250e^10k
divide by 250
1.12 = e^10k
ln 1.12 / 10 = .011 K = 0.11
Now I do not know what to do!
To find the time required for the population to double in size, we need to determine the value of t when N is twice its initial value.
Let's start by finding the initial value of N. We know that N = 250e^kt, and when t = 0 (initial time), we can substitute N = 250 and solve for k.
250 = 250e^0k
1 = e^0k
k = 0
Now that we know the value of k, we can rewrite the equation for N as N = 250e^0, which simplifies to N = 250.
To double the population, we need N to be 500 (twice the initial value of 250). So, we can set up the following equation:
500 = 250e^0.11t
To solve for t, we can divide both sides of the equation by 250 and take the natural logarithm (ln) of both sides:
ln(500/250) = ln(e^0.11t)
ln(2) = 0.11t
To isolate t, we divide both sides of the equation by 0.11:
t = ln(2) / 0.11
Using a calculator, we can find that ln(2) is approximately 0.693. Therefore:
t = 0.693 / 0.11
t ≈ 6.3
So, the estimated time required for the population to double in size is approximately 6.3 hours.