8.3 Density & Design Modeling

Density: Turning Volume Into Mass, People, or Cost

Standard GEO-G.MG.2 applies density — a quantity per unit of area or volume — to real modeling situations. The core relationship is simple, but Regents questions bury it inside a multi-step geometry problem, so the skill is really setup and unit tracking.

The Density Relationship

$\text{density} = \frac{\text{amount}}{\text{size}}$

The "amount" is mass, population, or another count; the "size" is area or volume computed from geometry. Rearranged, amount = density × size. Read the wording carefully: if the question gives the amount and the size and asks for the rate, you divide; if it gives the rate and one measurement and asks for the total, you multiply. Deciding which operation the sentence calls for is half the battle on these items.

The exam almost always makes you compute the geometry first, then multiply or divide by the density. Note the two flavors: volume density (mass per cubic unit, the most common) and areal density such as population per square mile or paint coverage per square foot, which pairs with an area rather than a volume. Match the density's units to the geometric quantity you computed — a per-area rate multiplies an area, and a per-volume rate multiplies a volume. Capacity questions are just volume questions phrased around how much a container holds.

Worked Examples

Direct mass. A solid has volume 20 cm³ and density 2.5 g/cm³. Mass = density × volume = 2.5 × 20 = 50 grams. Track the units: (g/cm³)(cm³) = grams, so the setup is correct.

Cylinder to weight. A cylindrical container has radius 2 in and height 10 in, filled with material weighing 0.5 lb/in³. First the volume: V = πr²h = π(2²)(10) = 40π in³. Then weight = 0.5 × 40π = 20π pounds. Notice you keep π exact all the way through.

Population density. A town has population 125,000 and land area 50 mi². Density = 125,000 ÷ 50 = 2,500 people per square mile. Here you divide, because density is being found, not used. If a follow-up asks how many people a neighboring 8 mi² district holds at the same density, you multiply: 2,500 × 8 = 20,000 people.

Composite award. A glass award is a rectangular prism (10 × 5 × 4 cm) with a rectangular pyramid on top (same 10 × 5 base, height 6 cm), glass density 2.4 g/cm³. Prism volume = 10·5·4 = 200 cm³; pyramid volume = ⅓(10·5)(6) = ⅓(50)(6) = 100 cm³; total = 300 cm³. Mass = 2.4 × 300 = 720 grams. Composite geometry and density combine in one Part III/IV problem.

Unit Analysis — Your Error Check

Unit analysis (treating units as algebra) catches most mistakes. If you want grams and you have g/cm³, you must multiply by cm³ so the cubic centimeters cancel. If a question mixes units — density in g/cm³ but a dimension in millimeters — convert before multiplying. The reference sheet lists conversions (1 inch = 2.54 cm; 1 mile = 5,280 feet; 1 pound = 16 ounces), so use them rather than guessing.

Conversion example. A metal bar is a rectangular prism 40 mm by 20 mm by 100 mm with density 7.8 g/cm³. Because density uses cm³, convert first: 40 mm = 4 cm, 20 mm = 2 cm, 100 mm = 10 cm. Volume = 4 × 2 × 10 = 80 cm³, so mass = 7.8 × 80 = 624 grams. Skipping the conversion and multiplying millimeters would inflate the volume by a factor of 1,000 — a classic exam trap.

Design and Material Choice

GEO-G.MG.1 asks you to model a physical object with a geometric shape — a grain silo as a cylinder topped by a cone, a can as a cylinder, a tent as a triangular prism. Density then supports material decisions: given two materials with different densities and prices, compute the mass each design requires, multiply by cost per unit mass, and compare.

Material-choice example. A part has volume 50 cm³. Aluminum has density 2.7 g/cm³ at $0.02 per gram; steel has density 7.8 g/cm³ at $0.01 per gram. Aluminum mass = 2.7 × 50 = 135 g, costing 135 × $0.02 = $2.70. Steel mass = 7.8 × 50 = 390 g, costing 390 × $0.01 = $3.90. Aluminum is both lighter and cheaper here, though a real decision would also weigh strength. A lighter material (lower density) yields less mass for the same volume, which can mean lower shipping cost. On the exam, always state which quantity you are optimizing (mass, cost, or capacity) so your comparison answers the question actually asked.