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Density (area-based and volume-based)

20 practice questions 2 video lessons Theory + worked examples

Density (Area- and Volume-Based)

California Geometry • Standard G-MG.2 • Modelling with Geometry

Density (Area- and Volume-Based) is a topic in Modelling with Geometry in the California Common Core State Standards. It is aligned to Standard G-MG.2, which requires students to apply concepts of density based on area and volume in modeling situations.

Density expresses an amount per unit of area or volume — population per square mile, or mass per cubic centimeter.

California Geometry › Modelling with Geometry › Density (Area- and Volume-Based)  —  Standard G-MG.2

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Theory

Density is an amount per unit of area or volume:
  • Area density: \(D=\dfrac{\text{quantity}}{\text{area}}\) (e.g. people per square mile).
  • Volume (mass) density: \(D=\dfrac{\text{mass}}{\text{volume}}\) (e.g. grams per cubic centimeter).
Density has compound units — per square mile, per cubic centimeter — so always carry the units.
Area density: population per area Population density is the number of people divided by the land area they occupy. population density = people ÷ area one region
Population density is people divided by area.
Density Density Density area density = quantity / area volume density (mass) = mass / volume population density = people / area
Area-based and volume-based density.

The density relationships:

\[D=\dfrac{\text{quantity}}{\text{area}}\quad\text{or}\quad D=\dfrac{\text{mass}}{\text{volume}}\]
density is a quantity divided by area, or mass divided by volume
Rearrange to get quantity \(=\) density \(\times\) area (or volume).

How to work with density

  1. Identify whether it is per area or per volume.
  2. Divide the quantity by the area or volume.
  3. Attach the compound units.
  4. To find a quantity, multiply density by the area or volume.
Example 1 — Population density
A town has \(60{,}000\) people on \(20\) square miles. Find its population density.
Solution

Density is people per unit area.

\(D\)\(=\)\(\dfrac{60{,}000}{20}\)
\(=\)\(3000\ \text{people/mi}^2\)
the density is 3000 people per square mile
Example 2 — Mass density
A metal block has mass \(386\) g and volume \(20\ \text{cm}^3\). Find its density.
Solution

Density is mass per unit volume.

\(D\)\(=\)\(\dfrac{386}{20}\)
\(=\)\(19.3\ \text{g/cm}^3\)
the density is 19.3 grams per cubic centimeter
Example 3 — Solve for a quantity
A crowd has a density of \(4\) people per square meter over \(150\ \text{m}^2\). How many people is that?
Solution

Multiply density by area.

\(N\)\(=\)\(4\times 150\)
\(=\)\(600\ \text{people}\)
there are 600 people
Example 4 — Compare densities
City A: \(9000\) people on \(3\ \text{mi}^2\). City B: \(9000\) on \(4.5\ \text{mi}^2\). Which is denser?
Solution

Compute each density.

\(D_A\)\(=\)\(\dfrac{9000}{3}=3000\)
\(D_B\)\(=\)\(\dfrac{9000}{4.5}=2000\)

City A is denser (\(3000>2000\)).

city A is denser at 3000 versus 2000 per square mile

Common pitfalls

Use area for 2D density, volume for 3D density — don't mix square and cubic units.
Keep the compound units, such as people/mi\(^2\).
To find the total, multiply density by area or volume; don't divide again.

Frequently asked questions

What is density?

An amount per unit of area or volume, such as people per square mile or grams per cubic centimeter.

What is population density?

The number of people divided by the land area, e.g. people per square mile.

What is mass density?

Mass divided by volume, e.g. grams per cubic centimeter.

How do you find a total from a density?

Multiply the density by the area or volume.

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