USA - Geometry
Modelling with geometry
Density (area-based and volume-based)
20 practice questions
2 video lessons
Theory + worked examples
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.
Population density is people divided by area.
Area-based and volume-based density.
The density relationships:
\[D=\dfrac{\text{quantity}}{\text{area}}\quad\text{or}\quad D=\dfrac{\text{mass}}{\text{volume}}\]
Rearrange to get quantity \(=\) density \(\times\) area (or volume).
How to work with density
- Identify whether it is per area or per volume.
- Divide the quantity by the area or volume.
- Attach the compound units.
- 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.
| \(D\) | \(=\) | \(\dfrac{60{,}000}{20}\) |
| \(=\) | \(3000\ \text{people/mi}^2\) |
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.
| \(D\) | \(=\) | \(\dfrac{386}{20}\) |
| \(=\) | \(19.3\ \text{g/cm}^3\) |
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.
| \(N\) | \(=\) | \(4\times 150\) |
| \(=\) | \(600\ \text{people}\) |
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 (\(3000>2000\)).
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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