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 estimate the time required for the population to double in size, we need to solve for t when N is equal to twice the initial value of N. In this case, the initial value of N is 250.

Let's start by setting up the equation for N when the population doubles:

2 * 250 = 250 * e^(0.11 * t)

We can simplify this equation by canceling out the common factor of 250:

2 = e^(0.11 * t)

To solve for t, we need to take the natural logarithm (ln) of both sides of the equation, as the natural logarithm is the inverse of the exponential function e^x:

ln(2) = ln(e^(0.11 * t))

Using the property of logarithms that ln(e^x) is equal to x, we have:

ln(2) = 0.11 * t

Now, to isolate t, we divide both sides of the equation by 0.11:

t = ln(2) / 0.11

Using a calculator, you can find that ln(2) is approximately 0.6931. Therefore:

t ≈ 0.6931 / 0.11

t ≈ 6.301 hours

So, it would take approximately 6.301 hours for the population to double in size.