- What is the relationship between the length of a square and the area of a square (options are linear, proportional, or neither)?

- What is the relationship between the E and m in the equation E=mc2?

Hello, 2y4life,

When we use the terms linear and proportional, we’re talking about how one thing changes with respect to another. It’s very useful to graph them to see what is going on. If the graph makes a straight line, we say there is a “linear” relationship between the variables. If the graph is curved instead, we explain it as some type of “proportionality.”

For example if you consider the relationship between “the length of a side of a square (s)” to its “area (A)” and graph them (with s on the x-axis and A on the y-axis), you don’t get a straight line. Instead, you get a curve called a parabola. Why? The equation for area is

A = s^2

The exponent, 2, on s is the reason. Therefore, we say “A is proportional to the square of s.”

Now let’s look at E = mc^2. We’re asked to describe the relationship between E and m. We see that there is an exponent of 2 on “c,” but it’s not on either of the variables we’re comparing. “c” is just a constant, and even when we square a constant, the result is still just another constant. So we could rewrite the equation into a more general form and simply say

E = m*(some constant)

If we graphed E vs m, with m on the x-axis and E on the y-axis, the graph would be a straight line. (The slope of the line would be the c^2 constant.) Therefore, the relationship between E and m is “linear.”

Hope that helps.