This problem involves applying density in geometric modeling to find the mass of a thin plate. Areal mass density represents mass per unit area—in this case, 0.4 grams per square centimeter. The relevant geometric measure is the plate's area, calculated as length times width: 10 × 6 square centimeters. To find the plate's mass, multiply the areal density by the area: 0.4 × (10 × 6). This calculation works because areal density times area equals total mass. A common mistake is adding dimensions (10 + 6) instead of multiplying them, which would give perimeter rather than area. Always check units for consistency: g/cm² × cm² = g, confirming you're calculating mass correctly from an area-based density.

This problem involves applying density in geometric modeling to find the total population in a circular district. Population density measures the number of people per unit area—in this case, 1,200 people per square mile. The relevant geometric measure is the circle's area, calculated using the formula π × radius²: π(3²) square miles. To find the total population, multiply the density by the area: 1,200 × π(3²). This calculation works because density times area equals the total quantity. A common mistake is using circumference (2πr) instead of area (πr²), which would give a linear measure rather than the needed area. Always verify units: people/square mile × square miles = people, confirming you're finding population.

This problem involves applying density in geometric modeling to find the total energy in a rectangular tank. Energy density represents the amount of energy per unit volume—here, 18 kJ per cubic meter. The relevant geometric measure is the tank's volume, calculated as length × width × height: 4 × 3 × 2 cubic meters. To find total energy, multiply the energy density by the volume: 18 × (4 × 3 × 2). This multiplication gives the total because density times volume equals the total quantity. A common error is adding the dimensions (4 + 3 + 2) instead of multiplying, which fails to calculate volume correctly. When working with density problems, verify your units: kJ/m³ × m³ = kJ, confirming you're finding total energy.

This problem requires applying density in geometric modeling to find the mass of a cylindrical rod. Density measures mass per unit volume—in this case, 7.8 grams per cubic centimeter. The relevant geometric measure is the cylinder's volume, given by the formula π × radius² × height: π(2²)(15) cubic centimeters. To find the rod's mass, multiply the density by the volume: 7.8 × π(2²)(15). This calculation works because density times volume equals total mass. A common mistake is using the wrong volume formula, such as 2 × 15 (treating it as a rectangle), which ignores the circular cross-section. Always verify units match: g/cm³ × cm³ = g, confirming you're calculating mass correctly.

This problem requires understanding how to interpret density in geometric modeling contexts. Mass density is defined as mass per unit volume, meaning how much mass exists in each cubic meter of space. For a density of 30 kg/m³, this means there are 30 kilograms of foam in every cubic meter of the material's volume. The correct interpretation recognizes that density relates to three-dimensional space (volume), not two-dimensional surface area or one-dimensional edge length. A common misconception is thinking density relates to surface area (square meters) rather than volume (cubic meters). When interpreting density statements, always check the units: kg/m³ clearly indicates mass per unit volume, not per unit area or length.

This problem requires applying density in geometric modeling to find the mass of a spherical ball. Density measures mass per unit volume—here, 1.1 grams per cubic centimeter. The relevant geometric measure is the sphere's volume, given by the formula (4/3)π × radius³: (4/3)π(5³) cubic centimeters. To find the ball's mass, multiply the density by the volume: 1.1 × (4/3)π(5³). This multiplication gives total mass because density times volume equals mass. A common error is using the surface area formula 4π(5²) instead of volume, which would give a two-dimensional measure when three-dimensional volume is needed. Verify units match: g/cm³ × cm³ = g, ensuring you're calculating mass from volume-based density.

This problem requires applying density in geometric modeling to find the mass of water in an aquarium. Water density represents mass per unit volume—here, 1.0 gram per cubic centimeter. The relevant geometric measure is the water's volume, calculated as base area times height: 50 × 30 × 40 cubic centimeters. To find the water's mass, multiply the density by the volume: volume (50 × 30 × 40) times density (1.0). This multiplication gives total mass because density times volume equals mass. A common error is using only the base area (50 × 30) without including height, which would give a two-dimensional measure instead of volume. Check units to confirm: g/cm³ × cm³ = g, verifying you're calculating mass.

This problem requires applying density in geometric modeling to find the total number of people in a rectangular park. Density is a measure of how much of something exists per unit of space—in this case, 2,400 people per square mile. The relevant geometric measure is the park's area, which equals length times width: 1.5 × 0.8 square miles. To find the total number of people, we multiply the density by the area: 2,400 × (1.5 × 0.8). This gives us the total because density times area equals the total amount. A common misconception is adding the dimensions (1.5 + 0.8) instead of multiplying them, which would give perimeter-based thinking rather than area. To avoid errors, always check that your units work out: people/square mile × square miles = people.

This problem tests understanding of how density applies in geometric modeling by identifying an incorrect claim. Energy density of 250 J/m³ means 250 joules of energy per cubic meter of volume. The correct way to find total energy is to multiply this density by the volume: 250 × 12 joules. The unit J/m³ indeed means joules per cubic meter, and doubling the volume would double the total energy (since total = density × volume). However, multiplying density by surface area is incorrect because density relates to volume, not area—this mixing of dimensions (3D density with 2D area) produces meaningless results. When using density, always ensure dimensional consistency: volume-based density requires multiplication by volume, not area.