12.6 Density & Geometric Modeling

Density & Geometric Modeling

The DRC Level A specification includes a modeling-with-geometry standard whose whole content is density: "apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot)." It is a small standard with a large payoff, because it combines the geometry of Chapters 11 and 12 with the rate reasoning of Chapter 4.

What Density Is

Density is an amount of something divided by the space it occupies.

$\text{Density} = \frac{\text{amount}}{\text{space}}$

The unit tells you which kind of space:

Because density is a rate, everything from Section 4.1 applies: it can be scaled up, scaled down, and rearranged.

Area-Based Density

Population density

$\text{Population density} = \frac{\text{number of people}}{\text{land area}}$

A county has 148,000 residents and covers 640 square miles. What is its population density?

$148{,}000 \div 640 = \mathbf{231.25 \text{ people per square mile}}$

Working backward is the more common TABE version:

A city has a population density of 4,200 people per square mile and covers 38 square miles. What is its population?

$4{,}200 \times 38 = \mathbf{159{,}600 \text{ people}}$

The unit reasoning confirms the operation: $\frac{\text{people}}{\text{mi}^2} \times \text{mi}^2 = \text{people}$. Square miles cancel and people remain.

Other area densities

Coverage as a reciprocal. Paint is sold as "350 square feet per gallon," which is area per amount — the reciprocal of density. To paint 1,400 sq ft: $1{,}400 \div 350 = \mathbf{4}$ gallons. Divide when the rate is given as space-per-amount; multiply when it is amount-per-space. Checking units resolves which every time.

Volume-Based Density

Mass density

$\text{Density} = \frac{\text{mass}}{\text{volume}} \qquad \text{so} \qquad \text{mass} = \text{density} \times \text{volume}$

Worked example. A concrete footing measures 4 ft × 3 ft × 2 ft. At 150 lb/ft³, what does it weigh?

Volume: $4 \times 3 \times 2 = 24$ cu ft. Weight: $24 \times 150 = \mathbf{3{,}600}$ pounds, or 1.8 tons.

Energy density

The standard's own second example is BTUs per cubic foot, the measure used for fuel gas.

Natural gas delivers about 1,030 BTU per cubic foot. A furnace needs 82,400 BTU per hour. How many cubic feet per hour does it consume?

$82{,}400 \div 1{,}030 = \mathbf{80 \text{ cubic feet per hour}}$

Units again: $\text{BTU} \div \frac{\text{BTU}}{\text{ft}^3} = \text{ft}^3$.

Modeling With Geometry

Geometric modeling means approximating a messy real object with a clean solid so its formulas apply.

The discipline the standard expects: state your assumption. "Modeling the trunk as a cylinder of uniform 18-inch diameter" is a complete answer; silently treating a tapered trunk as a cylinder is not.

Full worked model. A landscaper needs to order topsoil for a bed measuring 40 ft by 18 ft, spread 4 inches deep. Topsoil weighs about 90 lb per cubic foot and is sold by the cubic yard. How many cubic yards, and what will it weigh?

1. Model: a rectangular prism, treating the depth as uniform.
2. Consistent units: 4 inches $= \frac{4}{12} = \frac{1}{3}$ ft.
3. Volume: $40 \times 18 \times \tfrac{1}{3} = 240$ cu ft.
4. Convert: $240 \div 27 \approx \mathbf{8.9}$ cubic yards — order 9.
5. Weight: $240 \times 90 = \mathbf{21{,}600}$ pounds, about 10.8 tons.
6. Stated assumption: uniform depth, no settling or compaction allowance.

The Unit Check, One More Time

Every density problem is solved correctly by asking what unit the answer should carry.

If the units of your result do not match the units the question asks for, you performed the wrong operation — regardless of how sensible the number looks.