Question

A population of bacteria is initially 500. After two hours the population is 250. If this rate of decay continues, find the exponential function that represents the size of the bacteria population after t hours. Write your answer in the form f(t)=a(b)t. If you need to round any decimals, round to four decimal places.

Answer 1

500(1/2)^t/2

Explanation:

Use the exponential function f(x)=a(b)ct. The initial population, 500, gives the the point (0,500) and leads to coefficient of the exponential function, a=500.

f(t)=2000(b)ct

After 2 hours, the population has decreased by half. This means the common ratio is one-half, b=12. Because it takes 2 hours for the population to be cut in half, we know ct=1 when t=2, therefore c=1t and c=12. This gives the equation:

f(t)f(t)=500(12)12(t)=500(12)t2

Alternate Solution

The situation shows there are two points (0,500) and (2,250). Plugging the first point in, you solve for a=500 as follows:

500a=a(b)0=500

The decay coefficient, b, can be determined by substituting in the value for a and the point (2,250) and the solving as follows:

25025050012(12)12b=500(b)2=b2=b2=b≈0.7071

This gives the final exponential equation:

f(t)=500(0.7071)t