<em><u>The polynomial that represents the difference between the cost of distributing by train and the cost of distributing by truck is:</u></em>

<em><u>Solution:</u></em>
<em><u>The cost of distributing by train can be modeled as:</u></em>

<em><u>The cost of distributing by truck can be modeled as:</u></em>

where x is the number of tons of product distributed
To find: Difference between the cost of distributing by train and the cost of distributing by truck
Difference = cost of distributing by train - cost of distributing by truck
Let f(x) = difference in cost between trains & trucks



Thus polynomial that represents the difference between the cost of distributing by train and the cost of distributing by truck is found
If we graph the given points then they appear to be in the shape of cubic function so we can use standard formula of cubic function which is

Plug the given points to get four equations
like first point (3,6) so plug x=3 and y=6 into above equation, we get:

or
6=27a+9b+3c+d
Same way by plugging other points form table, we get total four equations as follows:
6=27a+9b+3c+d, 11=125a+25b+5c+d, 11=64a+16b+4c+d, 21=343a+49b+7c+d
Now we just need to solve these equations for a, b, c and d.
Solving them to get values of a, b, c and d, we get:

Now plug them into

So the required equation will be:

Hence required equation in function notation can be written as

Now to prove that above function is correct, we just graph the given points from table and the obtained function.
After graphing the given points from table and the obtained function, We see that points lie on the graph of 
Which proves that our equation is correct.
Basically the line is 180° in total.
If one side is 129°
then the other side is 180 - 129
which means x = 51°