How Is the Constant A Determined in Janine's Charcoal Absorption Data Analysis?

[amanda_]

Homework Statement

Janine, a science student, measured the volume of carbon dioxide gas that can be absorbed by one cubic centimeter of charcoal at pressure x. She made the following four sets of measurements:

(120, 3.1)
(340, 5.5)
(534, 7.1)
(698, 8.3)

Janine finds that the volume, f(x), to be a function of pressure, x, and decides to use the formula discussed earlier in this project to fit her data. Recall that the formula is:

f(x) = A + B(x – xo) + C(x – xo)(x – x1) + D(x – xo)(x – x1)(x – x2)

where A, B, C, and D are constants, while xo, x1, x2 are the x-coordinates of three of the data points given.

When we use Janine’s data, what would the value of A be?

The Attempt at a Solution

f(x)=A+B(x-120)+C(x-120)(x-340)+D(x-120)(x-340)(x-534)

That's how I set it up, but I got lost after that. Do I need to factor everything else? I'm not sure how to find A if nothing else is given.

 

[HallsofIvy]

Now, put in values for x and f(x).
Having chosen the first three data points to give [itex]x_0[/itex], [itex]x_1[/itex], and [itex]x_2[/itex], It is easy to see that
f(120)= 3.1= A+B(120-120)+C(120-120)(120-340)+D(120-120)(120-340)(120-534)= A since all other terms have a factor of 120- 120= 0. A= 3.1

f(340)= 5.5= A+B(340-120)+C(340-120)(x-340)+D(340-120)(340-340)(340-534)= A+ 120B because all other terms have a factor of 340- 340= 0. A+ 120B= 5.5. Since you already know A, it is easy to solve for B.

f(534)= 7.1= A+B(534-120)+C(534-120)(534-340)+D(534-120)(534-340)(534-534)= A+ 114B+ (414)(194)C because the last term has a factor of 534- 535= 0. A+ 114B+ 80316C= 7.1. Since you already know A and B, it is easy to solve for C.

Finally, f(698)= 8.3= A+ B(698- 120)+ C(698- 120)(698- 340)+ D(698- 120)(698- 340)(698- 534)= A+ 578B+ (578)(358)C+ (578)(358)(164)D= A+ 578B+ 206924C+ 33935536D. Since you already know A, B, and C, it is easy to solve for D.

F