Linear cost function

[mschlach]

I need help with figuring out this problem please.
The cost function for wild crayfish was estimated by Bell to be a function C(x), where x is the number of millions of pounds of crayfish caught and C is the cost in millions of dollars. Two points that are on the graph are (x, C) = (8, 0.157) and (x, C) = (10, 0.190). Using this information and assuming a linear model, determine a cost function.

 

[Ishuda]

I need help with figuring out this problem please.
The cost function for wild crayfish was estimated by Bell to be a function C(x), where x is the number of millions of pounds of crayfish caught and C is the cost in millions of dollars. Two points that are on the graph are (x, C) = (8, 0.157) and (x, C) = (10, 0.190). Using this information and assuming a linear model, determine a cost function.

Using a formula for two given points you can write

\(\displaystyle y\, =\, \dfrac{y_1 (x \,-\, x_0) \,- \,y_0 (x \,- \,x_1)}{x_1\, -\, x_0}\)

and then put it in the form you wish [point-slope, intercepts, etc.].

 

[mschlach]

Using a formula for two given points you can write

\(\displaystyle y\, =\, \dfrac{y_1 (x \,-\, x_0) \,- \,y_0 (x \,- \,x_1)}{x_1\, -\, x_0}\)

and then put it in the form you wish [point-slope, intercepts, etc.].

Thanks a ton! I don't understand why there are so many variables though because I only have four different numbers.

 

[HallsofIvy]

"x" and "y" are the true variables and will remain as the letters "x" and "y". The four subscripted letters, \(\displaystyle x_0\), \(\displaystyle y_0\), \(\displaystyle x_1\), and \(\displaystyle y_1\) are the data you are given:
"(x, C) = (8, 0.157)" is \(\displaystyle x_0= 8\), \(\displaystyle y_0= 0.157\), and "(x, C) = (10, 0.190)" is \(\displaystyle x_1= 10\), \(\displaystyle y_1= 0.190\). Feel free to replace the letter "y" with "C".

Another way to do this: Any linear function can be written as "y= ax+ b" or, since your data is give in terms of "x" and "C", C= ax+ b, for some constants a and b.

You are told that "(x, C)= (8, 0.157)" so 0.157= 8a+ b. You are told that "(x, C)= (10, 0.190)" so 0.190= 10a+ b.


Solve the two equations 0.157= 8a+ b and 0.190= 10a+ b for a and b.