How do we find height when given density, mass and radius

Answer 1

One must assume that an object with both height and radius must be either a cylinder or a cone. There isn't enough information to distinguish which it is, but the idea for how to solve for height, given the radius and density is the same.

Density is Mass/Volume. We'll assume g/cm^3, or g/cc whre cc is Cubic Centimeter, cm^3.

Cylinder:

The volume of a cylinder is πr²h, where h is the height and r the radius. If we are given the density, radius, and mass, we can first calculate the volume, V, of the cylinder by:

Density(D) = m/V

V = m/D

Then we can say V=πr²h

Since we are given r, the radius, we can now rearrange to find height, h:

h = V/πr²

Since volume, V, was defined as m, mass, over the density, D, we can put everything into one equation:

h = (m/D)/(πr²)

Since m/D = V, we can also write:

h =V/(πr²)

We know the mass,m, density, D, and radius, r, and can look up the value of pi, π, to however many decimal points we want. Plug those in to determine height, h. Be sure to check that all are in consistent units of measure.

Cone:

The volume of a cone is (1/3)πr²h. So take this throught the same analysis as above, but add (1/3) to the expression:

V = (1/3)πr²h

V = m/D

m/D = (1/3)πr²h

h = (m/D)/((1/3)πr²)

h = 3m/Dπr²

Since m/D = V, we can also write:

h = 3V/πr²