Density and Other Derived Units

9.4: Density and Other Derived Units. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 4 of 73 parts Practice Flashcards

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Density as a quotient

Objective: Calculate density from mass and volume.

Prerequisite skills: volume formulas, unit conversions, and division. Density is mass per unit volume. A sample with twice the mass and twice the volume has the same density. Keep the units as a quotient, such as grams per cubic centimeter. Common error: dividing volume by mass. Restrictions and interpretation: Match the mass and volume units to the requested density; density is an intensive ratio.

Formulas

$ρ=m/V, V>0$

Worked examples

Try itA sample has mass 575 grams and volume 25 cubic centimeters. What is its density in g/cm³?Show answer

Density=m/V=575/25=23 grams per cubic centimeter. The positive volume makes the ratio defined. Therefore the requested answer is 23.

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Density as a quotient

Objective: Calculate density from mass and volume.

Formulas

$ρ=m/V, V>0$

Worked examples

Try itA sample has mass 575 grams and volume 25 cubic centimeters. What is its density in g/cm³?Show answer

Density=m/V=575/25=23 grams per cubic centimeter. The positive volume makes the ratio defined. Therefore the requested answer is 23.

Derived-unit modeling

Objective: Use a derived unit to recover a total amount.

Rearranging density=mass/volume gives mass=density×volume. Other derived rates work the same way: population density in people/km² times area gives people, and dollars/kWh times energy in kWh gives dollars. Keep the denominator units matched before multiplying. Common error: dividing by a per-unit rate when it should multiply the quantity. Restrictions and interpretation: Match the mass and volume units to the requested density; density is an intensive ratio.

Formulas

\[m=ρV; population=(people/area)×area; cost=($/kWh)×kWh\]

Worked examples

Try itElectricity costs $0.20 per kilowatt-hour. What is the cost in dollars of 250 kilowatt-hours, with no other charges?Show answer

Multiply the derived rate by energy: ($0.20/kWh)(250 kWh)=$50. The kWh units cancel, leaving dollars. Therefore the requested answer is 50.

Geometric density models

Objective: Combine a volume formula with density and unit consistency.

For a uniform solid, determine its volume from dimensions before using its density. A radius or diameter must be interpreted correctly. If a question asks for the density itself, divide mass by the computed volume. Common error: using surface area instead of volume in a density model. Restrictions and interpretation: Match the mass and volume units to the requested density; density is an intensive ratio. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

rectangular-solid mass=ρ·lwh

Worked examples

Try itA uniform rectangular block is 25 cm by 3 cm by 4 cm and has density 2 g/cm³. What is its mass in grams?Show answer

The volume is 25×3×4=300 cm³. Multiply by density: m=2(300)=600 grams. Surface area would have units cm² and cannot be used here. Therefore the requested answer is 600.

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