Question

Is density exponentially proportional to volume or inversely?

Answer 1

An exponential relationship takes the form y = k(a^x), where a and b are constants. Often the value of a turns out to be Euler's number, e = 2.718281828..., because it has the special property that d/dx(e^x) = e^x. Exponential growth might model how the population of a bacterium which divides every 5 seconds increases, and exponential decay might model how the mass of a radioactive isotope sample decreases with time.

Inverse proportion takes the form

An inverse proportion takes the form y = k/x, for a constant k. This kind of relationship might model how the time taken to complete a job varies as the number of workers varies, for example.

We know the formula for density is mass over volume (d = m/v). We therefore see that when mass is considered an arbitrary constant, substitute y = d, k = m and x = v, and we have our inverse proportion.

Intuitively this makes sense. Double the volume for the same mass and the density halves. Quarter the volume for the same mass and the density quadruples.

I hope this helps you :)