SOLUTION: Hi I am having a lot of difficulty forming an equation for c in terms of t from the following table t 1 1.5 2 2.5 3 5 c 50 60 70 80 90 130 could you please show your

Question 1019231: Hi
I am having a lot of difficulty forming an equation for c in terms of t from the following table
t 1 1.5 2 2.5 3 5
c 50 60 70 80 90 130
could you please show your workings as to how you arrived at the equation.
thank you
Found 3 solutions by Fombitz, Theo, rothauserc:
Answer by Fombitz(32388)   (Show Source): You can put this solution on YOUR website!
I fit the equation using EXCEL.
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If you know it's a straight line, you can determine the equation using any two points.
First find the slope.

Now use any point and the slope using the point-slope form of a line,

This was done using point 1 and point 6.
You can verify that each of the other points is also on the line.
Example point 2,

(,)
You would need to do this for the remaining points to verify that they are also on the line.
Or you could generate six lines and show that the slope and y-intercept are identical.
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
it appears that for every change in the value of x, there is a corresponding change in the value of y.

for example:

x from 1 to 1.5 is a change of .5
corresponding change in y is from 50 to 60 which is a change of 10.

following table illustrates:

x y change in x change in y ratio of change in y divided by change in x 1 50 1.5 60 .5 10 10/.5 = 20 2.0 70 .5 10 10/.5 = 20 2.5 80 .5 10 10/.5 = 20 3 90 .5 10 10/.5 = 20 5 130 2.0 40 40/2.0 = 20

the ratio of the change in y divided by the change in x is constant.

this is equivalent to the slope in a straight line.

the general equation of the slope intercept form of a straight line is:

y = mx + b

m is the slope
b is the y-intercept.

based on this data, the slope is determined to be 20.

the equation therefore becomes y = 20x + b

to find the value of b, just replace x and y with their respective values from one of the point pairs in the table.

we'll use the point (1,50)
this means the value of x is 1 when the value of y is 50.

the equation of y = 20x + b becomes 50 = 20*1 + b

solve for b to get b = 50 - 20 = 30.

the slope intercept form of the equation becomes y = 20x + 30.

the graph of that equation looks like this:

i didn't show the corresponding y values for x = 1.5 and 2.5, but you can easily determine that they will be y = 60 and 80 from the table above, as well as pin point the corresponding y value by inserting an imaginary vertical line at x + 1.5 and 2.5 and seeing that it will intersect the line of the equation at those indicated values of y.

the procedures for determining the algebraic relationship can get tricky.

in this case, i just notices that the value of y was increasing by 10 when the corresponding value of x was increasing by .5 which indicated the relationship described above.

Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
we use linear regression for this problem
*********************************************
we want to determine the following formula
c(t) = a + bt
********************************************
we have N = 6 pairs of data, first we calculate
tc and t^2 for each pair
t c tc t^2
1 50 50 1
1.5 60 90 2.25
2 70 140 4
2.5 80 200 6.25
3 90 270 9
3.5 130 455 12.25
********************************************
summation of t's = 13.5
summation of c's = 480
summation of tc's = 1205
summation of t^2's = 34.75
********************************************
slope(b) = ((N * summation of tc's) - (summation of t's * summation of c's)) / ((N * summation of t^2's) - (summation of t's)^2)
:
b = ((6 * 1205) - (13.5 * 480)) / ((6 * 34.75) - (13.5)^2)
:
b = 28.571428571 approx 28.57
**********************************************
intercept(a) = (summation of the c's -(b*summation of t's)) / N
a = (480 - (28.57 * 13.5)) / 6
:
a = 15.7175 approx 15.72
**********************************************
equation for c is
c(t) = 28.57x + 15.72
*********************************************

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