Influence of each variable to the result of equation

[bibis3000]

Hi,

This is more about asking for reading material and general direction.
Lets say there is a party planned. We are expecting 5 guests and every guest is expected to drink two sodas. So we are planning for 5 x 2 = 10 sodas.
The day of the party comes and 8 thirsty guests show up, each of them drinking 4 sodas. 8 x 4 = 32 sodas. All in all, 32-10 = 22 more sodas were consumed than planned.
Now the question: what are the "influences" of the guest count and the guest thirst on greater total soda consumption?
I was thinking:
Guest count influence: (8-5) x 2 = 6
Guest thirst influence: 8 x (4-2) = 16
Total soda consumption increase: 6 + 16 = 22.
I would be grateful if anyone could tell me how these types of math problems are called and where could i read up on general approach when solving them.
In reality, the "total soda consumption" equation has 4 to 5 variables and looks like (Ax(B-C)+D)/E. I am trying to find out how each of the 5 variables influence the the difference between "planned" and "actual" consumption. I hope this makes any sense.

Have a nice day!

 

[bZNyQ7C2]

I might be missing the point altogether, but I see that a number of people have looked at this without commenting, so I'll take a shot.

I think in terms of geometry and linear algebra. The expected consumption I picture as a rectangle 5 units long and 2 units high. In a standard basis, the vertices would be (0,0), (5,0), (5,2) and (0,2). The actual consumption would then be another rectangle 8 units long and 4 units high. Its vertices would be (0,0), (8,0),(8,4) and (0,4). The mapping between these two rectangles would then be a simple linear transformation, a dilation, of which the matrix is

\(\displaystyle \left[ {\begin{array}{*{20}{c}} {1.6}&0\\ 0&2 \end{array}} \right]\)

This is a diagonal matrix, so the two "influences" seem to be orthogonal (mutually independent). The eigenvalues are 1.6 and 2 (pretty clearly) and a set of eigenvectors might be

\(\displaystyle \left[ {\begin{array}{*{20}{c}} 5\\ 0 \end{array}} \right]{\rm{ and }}\left[ {\begin{array}{*{20}{c}} 0\\ 2 \end{array}} \right]\)

If the question is which factor or "influence" does more to increase the consumption, I suppose you could say it's the thirst of the guests, which is twice what was expected.

That's all I have to say, I hope it's helpful.

 

[HallsofIvy]

Your function here is simply the number of guests times the average number of sodas each guest drinks: f(x,y)= xy. The derivative with respect to x is \(\displaystyle f_x(x,y)= y\) and the derivative with respect to y is \(\displaystyle f_y(x,y)= x\). That is what bZNyQ7C2 is saying- if one of the variables is larger than the other, then the other variable "influences" the result more.

 

[bibis3000]

Ok, I think you got me on the right path. Thank you!