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

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1 Answer
Sep 19, 2017

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#