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[Guest]

could someone help me with please

Quantities x and y are believed to be related by a law of the form y = mnx. The values of x and corresponding values
of y are:
x 0.5 1.0 1.5 2.0 2.5 3.0
y 0.53 3.0 8.27 16.97 29.69 46.77
Draw graphically then verify the law and find the values of m and n.

[Guest]

y=mnx? That's a little odd. Whether "mn" represents one number or is a product of m and n that is a single number. This can be written as y= Ax with A= mn. The graph of that is a straight line passing through (0, 0) and with slope A= mn.

The first point given is (x, y)= (0.5,0.53). Putting that into y= mnx, we have 0.53=mn(0.5) so mn= 0.53/0.5= 1.06. The next point is (1.0, 3.0) so 3= mn. That is very different from 1.06 so these two points do NOT lie on a straight line through the origin or anywhere near such a line.

I am still concerned about the "mn". Is it possible that this was y= mx+n or y= mx^n? I haven't calculated for those but those might work.

[Guest]

Or possibly y= mn^x?

[Guest]

That's a good point- though it would have been better for the OP to have explicitly said so!!

If the formula is to be y= mn^x then we have two parameters to be determine, m and n. And we need two equations to do that. Here we have six, which is "overkill". It is quite possible that the answer will depend on which two data points we take- that the formula is NOT precisely "y= mn^x" for any m and n.

If I suspect such a situation, I am inclined to use the two data points farthest apart to make it as general as possible. Here, those are (0.5, 0.53) and (3.0, 46.77).

So we have [tex]0.53= mn^{0.5}[/tex] and [tex]46.77= mn^3[/tex]. We can eliminate "m" from the equations by dividing the second by the first: [tex]\frac{46.77}{0.53}= \frac{mn^{3}}{mn^{0.5}}[/tex] or [tex]88.24= n^{3- 0.5}= n^{2.5}[/tex]. Taking the 2.5 root of both sides, [tex]n= 35.30[/tex]. Then [tex]46.77= m(35.30^3)= 43980m[/tex] so [tex]m= \frac{46.77}{43980}= 0.001063[/tex].

Another idea is to use "least squares" to find a curve, y= mn^x the comes closest to all points without necessarily going through any of them.

[HallsofIvy]

On further thought I am inclined to think that original post really intended [tex]y= m^{nx}[/tex].