12In an experiment, a petri dish with a colony of bacteria is exposed to cold temperatures and then warmed again.

Find a quadratic model for the data in the table. Type your answer below. Show your work.

Time (hours)

0

1

2

3

4

5

6

Population (1000s)

5.1

3.03

1.72

1.17

1.38

2.35

4.08

Let the equation be : at^2 + bt + c = P,

where t = time (hrs),

P = population (1000's).

When t = 1, P = 3.03.

When t = 2, P = 1.72.

When t = 3, P = 1.17.

Substitute these into the equation to obtain these 3 simultaneous equations : a + b + c = 3.03

4a + 2b + c = 1.72

9a + 3b = c = 1.17

Solving gives:

a = 0.38,

b = - 2.45,

c = 5.1.

The equation is therefore,

P = 0.38t^2 - 2.45t + 5.1

Testing with t = 0 to 6 gives the population values as provided, so it seems to be a valid model.

At t = 9 hrs,

P = 0.38*9^2 - 2.45*9 + 5.1

= 13.83.