Question

Can I get one last help please? I truly don't understand it. Thanks!

1. Use the model from problem 12 to estimate the population of bacteria at 9 hours.
   Type your answer below. Show your work.

The picture is how I solved #12

Answer 1

13.83

Explanation:

So, the constant rate of change indicated by the diferences of the diferences tells us the generic equation here is a quadratic one. Quadratic equations have the generic formula:
`y=ax^2+bx+c`

The first thing we do is look at the point where x is equal to 0 `(0,5.1)` in order to find c:

`5.1=a*0^2+b*0+c`
`c=5.1`

Now, in order to find a and b we need to pick two points from the list and substitute them into the generic equation to get two linear equations. Lets choose `(1,3.03)` and `(6,4.08)`:
1) `3.03=a*1^2+b*1+5.1`
`-2.07=a+b`

2) `4.08=a*6^2+b*6+5.1`
`-1.02=36a+6b`

In order to solve a system like this we multiply or divide one of the equations by a set number so that one of the terms equals the same term in the other equation. Lets multiply the first equation by -6:
`-6*-2.07=-6*(a+b)`
`12.42=-6a-6b`

Now we add both equations
`12.42+(-1.02)=36a+6b-6a-6b`
`11.4=30a`
`a=11.4/30`
`a=0.38`

Stick that value into one of the equations:
`-2.07=0.38+b`
`b=-2.45`

So, our full equation is:
`y=0.38x^2-2.45x+5.1`

Now, in order to find the population at 9 hours, we just plug 9 for the value of x and find:
`y=0.38*9^2-2.45*9+5.1`
`y=13.83`