Using the slope of the graph, write an equation for this proportion

[F.B]

Im in gr.12 and my teacher gave us a dry lab to do. I have all the information. But there's a question that i can't answer. So can you please help me.

The purpose of this lab is to determine the relationship between the resistance, the length and the diameter of an electrical conductor.


This is the data. This table won't come out very good so please try your best to understand it. The data ends up crunched up together for some reason

Diameter (cm) Resistance 1/d^2
0.025 15.7 1600
0.04 6.1 625
0.075 1.9 177.7
0.125 0.6 64


The 1/d^2 is when the inverse motion becomes a straight line because we are doing proportionality statements.

Anyways this is the question i need help on.

When a straight line is achieved which is 1/d^2, write a proportionality statement relating R and d, for a constant length. Using the slope of the graph, write an equation for this proportion.

Can anyone help me with this question?

 

[Diane_]

If there's an indirect proportionality between, say, x and y^2, that means the relation can be written as

x = k/y^2

where k is a constant. That constant will be the slope of the line you get when you graph x against 1/y^2.

 

[F.B]

I can't use x=something. That would be sort of wrong i would have to use y=something because y is the resistance. So it would have to be y=kx^-2 or y=k/1/x^2.

 

[F.B]

I need help once again. I need to post all the data from my lab but its going to get all crunched up again.

sorry is its really hard to read the chart but i don't know how to fix it. Anyways there are 4 different lengths and at each length the resistance gets stronger and stronger. My question is that i have to create a proportionality statement between the length diameter and resistance. So can anyone please help me.

Length 100 140 170 220
Diameter (cm) Resistance
0.025 15.7 22.0 26.8 34.7
0.04 6.1 8.5 10.4 13.3
0.075 1.9 2.5 3.0 3.9
0.125 0.6 0.9 1.1 1.4

 

[Diane_]

First off, the 'x' and 'y' are just variables, and you're free to define them as you wish. If you find it easier to think of it as y = k/x^2, feel free. It won't change anything.

As for the other, I'm sorry, but I can't read the data well enough to help you. All I can do is offer some generic advice and hope it works.

The easiest way to handle this would be to isolate your variables. This would mean plotting resistance vs. length for wires of common diameter, then plotting resistance vs. diameter for wires of constant length. In this way, you can see the relationships between diameter and resistance and length and resistance without other factors intruding. Once you know those relationships, you can combine them into a single equation and test it against the totality of your data. The problem is that I don't know if you have the kind of data you can use for this analysis. From your description, it appears that you might, but I just can't tell.

If you don't, you can analyze it all at once, but it becomes harder. You'd need to plot your data on a three-dimensional grid, going for resistance vs. length and diameter. If you get the right powers on the length and diameter, then your results will be a straight line with the slope of that line being the constant of proportionality. The problem is that, while it's relatively easy to mess with the powers for a single variable until you get a straight line, it's geometrically harder as you add in independent variables. One also needs a decent knowledge of geometry to work in three dimensions. Fortunately, all you really need to work with (once you have the right powers) are linear functions, but as the whole point is to find the powers, I'm not sure how much help that will be.

Does that help at all?