The count in a bacteria culture was 400 after 10 minutes and 1100 after 40 minutes. Assuming the count grows exponentially,

Question

The count in a bacteria culture was 400 after 10 minutes and 1100 after 40 minutes. Assuming the count grows exponentially,

What was the initial size of the culture?    

Find the doubling period.    

Find the population after 105 minutes.    

When will the population reach 10000.    

You may enter the exact value or round to 2 decimal places.

Answer 1

There are two parameters we need to solve for to find the equation for an exponential function: a₀, the initial population or amount, and b, the base of the exponential function, that controls the growth or decay rate and determines whether the function models growth (b > 1) or decay (0 < b < 1). 2 given points allows us to solve for both, b first. The equation we get will be of the form P(t) = a₀b^t, where t is time (min) and P(t) is population

P(40): a₀b^40 = 1,100

P(10): a₀b^10 = 400

Divide the top equation by the bottom, left side ÷ left side , right side ÷ right side:

b^30 = 1.1/4 Take the 30th root of both sides to get b. Plug that back in to either equation to solve for a₀.

Once you have the equation, doubling time will be ln(2)/ln(b) (good formula to memorize for exponentials).

P(105) is simple to evaluate. Solving for the time the pop = 10,000 requires substituting that number for P(t) and solving the resulting exponential equation, which always requires taking a log of both sides (that's what gets a variable out of the exponent, since ln(b^t) = t lnb.