The Size and Distribution of Cities
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AP Human Geography › The Size and Distribution of Cities
A secondary-source excerpt discusses hexagonal trade areas in central place theory: hexagons are used to represent market areas that efficiently cover space without gaps or overlaps, assuming uniform terrain and equal travel cost in all directions. Which statement best reflects the excerpt’s explanation of why hexagons are used?
Hexagons indicate a primate city dominates the national economy and politics.
Hexagons are a simplified geometric tool to model efficient market coverage under idealized assumptions.
Hexagons are a real, observable shape of all metropolitan areas on maps, regardless of terrain.
Because hexagons ignore limitations, they should be used to draw exact service boundaries in planning.
Hexagons appear because every city’s population follows the rank-size rule exactly.
Explanation
In central place theory, hexagonal trade areas are theoretical shapes that efficiently cover space without overlaps or gaps, assuming uniform terrain and equal travel costs. Hexagons are chosen over circles because circles would leave unserved gaps or overlaps, while hexagons tile perfectly. This geometric tool models how market areas might form under idealized conditions to minimize travel for consumers. Choice C accurately reflects this as a simplified model for efficient coverage, not a literal map of cities. In reality, terrain like mountains or rivers distorts these shapes, but the concept aids in understanding spatial organization. The excerpt emphasizes assumptions, so hexagons are not used for exact planning but for theoretical insight.
A secondary source excerpt summarizes Christaller’s central place theory, explaining that settlements form an urban hierarchy in which higher-order centers provide specialized goods and services to larger hinterlands, while lower-order centers provide everyday goods to smaller market areas. In a region, Town X has a hospital, university, and major shopping mall; Town Y has a clinic and grocery stores; Town Z has only convenience stores. Which interpretation best matches central place theory?
Town X is a higher-order center because it provides higher-order services that require a larger market area and population threshold.
Town X is a primate city because it is the largest settlement, which central place theory says should dominate the entire country.
Central place theory proves real settlements always form perfect hierarchies, so Town Y cannot exist between X and Z.
Town Z is a higher-order center because convenience stores have the largest range and therefore serve the biggest hinterland.
Because the excerpt is a model, it cannot be used to interpret any real-world service patterns.
Explanation
Central place theory, developed by Walter Christaller, explains urban hierarchies where higher-order centers offer specialized services to larger areas, and lower-order ones provide basic goods to smaller zones. Town X, with a hospital, university, and mall, fits as a higher-order center due to these services' larger thresholds and ranges. Town Y offers mid-level services like clinics, and Town Z only basic ones, forming a hierarchy. Choice A correctly applies this by noting Town X's role in serving a bigger market. Choice B wrongly calls it a primate city, which is a national dominance pattern, not local hierarchy, and Choice C reverses the order by misunderstanding convenience stores as high-order.
In an idealized version of central place theory, market areas are often represented as hexagons to minimize overlap and gaps when serving consumers across a flat plain. A planning agency is using this model to propose evenly spaced service centers for emergency clinics. Which reasoning best supports the use of hexagonal trade areas in the model?
Hexagonal trade areas cannot be considered because the model has limitations, so geometric representations should never be used.
Hexagons are used because they approximate equal access while tessellating space without gaps, unlike circles.
Hexagons are used because rank-size requires six cities of equal size around each metropolis.
Hexagonal trade areas will appear exactly on the ground if planners adopt the model, regardless of terrain and road networks.
Hexagons are used because primate cities always have six subordinate cities arranged around them.
Explanation
In central place theory, hexagons are used to represent market areas because they can tessellate (fit together) without gaps or overlaps while approximating circular service areas. This ensures complete coverage of space while minimizing travel distances for consumers. Choice A correctly explains this geometric reasoning. Choice B incorrectly relates hexagons to rank-size rule. Choice C wrongly connects them to primate cities. Choice D treats the model too literally, ignoring real-world constraints. Choice E dismisses geometric models entirely when they serve as useful abstractions for planning purposes.
A country’s city-size distribution is being evaluated using the rank-size rule. The largest city has 10 million people. The next three cities have 6.5 million, 6.1 million, and 5.9 million. Which assessment is most accurate?
The distribution shows central place theory because cities of similar size indicate equal thresholds across the country.
The distribution fits rank-size because having several cities close in size is the definition of a primate city system.
The distribution deviates from rank-size because the second, third, and fourth cities are much larger than the $\frac{1}{n}$ expectation (5 million, 3.3 million, 2.5 million).
The distribution must be rank-size because all countries naturally produce $\frac{1}{n}$ city sizes over time.
Rank-size cannot be applied because it ignores limitations, so calculating expected sizes is meaningless.
Explanation
According to rank-size rule, with a 10 million person largest city, we expect: 2nd city = 5 million, 3rd = 3.3 million, 4th = 2.5 million. The actual cities (6.5, 6.1, and 5.9 million) are all much larger than predicted, showing a flatter distribution with several cities of similar size rather than the steep hierarchy predicted by rank-size. Choice A correctly identifies this deviation from the expected 1/n pattern. Choice B confuses this pattern with primate cities. Choice C treats rank-size as inevitable. Choice D misapplies central place theory. Choice E dismisses the analysis when the calculations clearly show meaningful deviation.
Christaller’s central place theory describes an idealized urban hierarchy in which higher-order centers provide specialized goods and services to larger market areas, while lower-order places provide everyday goods to smaller surrounding areas. In a region with many small towns, a few mid-sized cities, and one large metropolis with specialized hospitals, universities, and corporate headquarters, which statement best applies central place theory?
The region follows central place theory because specialized services cluster in higher-order centers that serve wider hinterlands.
The region must match Christaller’s ideal geometry exactly, since the model describes how settlements always distribute on real terrain.
Central place theory is unusable here because it ignores limitations such as physical geography and transportation networks, so no conclusions can be drawn.
The region follows a primate pattern because central place theory requires one city to dominate the entire country.
The region shows rank-size because central place theory predicts each city will be exactly $\frac{1}{n}$ the size of the largest.
Explanation
Central place theory explains urban hierarchies based on the range of goods and services offered. Higher-order centers (like the metropolis) provide specialized services (hospitals, universities, corporate headquarters) that require larger market areas, while lower-order places (small towns) provide everyday goods to smaller areas. Choice A correctly applies this concept to the described region. Choice B incorrectly conflates central place theory with primate cities. Choice C confuses it with rank-size rule. Choice D wrongly treats the model as prescriptive rather than descriptive. Choice E dismisses the theory entirely when it can clearly provide useful insights about service distribution.
Urban geographers note that in many countries the rank-size rule approximates the size distribution of cities: the population of the $n$th-largest city is about $\frac{1}{n}$ of the largest city. In Country X, the largest city has 8 million people and the next four cities have populations close to 4.1 million, 2.7 million, 2.0 million, and 1.6 million. Based on the rank-size rule, which interpretation best fits this pattern?
Country X closely follows the rank-size rule because the city sizes roughly match the $\frac{1}{n}$ relationship with the largest city.
Country X follows a primate city pattern because the second-largest city is half the size of the largest, which is the definition of rank-size.
Country X exhibits a central place hierarchy because the largest city is exactly twice the size of the second city.
Country X’s city distribution cannot be assessed with rank-size because the rule ignores any limitations such as migration, planning policy, or uneven development.
Country X must follow the rank-size rule perfectly in all regions since models describe how cities actually form.
Explanation
The rank-size rule states that the nth-largest city should have approximately 1/n the population of the largest city. For Country X with an 8 million person largest city, we would expect: 2nd city = 8/2 = 4 million, 3rd = 8/3 = 2.67 million, 4th = 8/4 = 2 million, and 5th = 8/5 = 1.6 million. The actual populations (4.1, 2.7, 2.0, and 1.6 million) closely match these predictions. Choice B correctly identifies this pattern. Choice A incorrectly defines rank-size as the second city being half the size of the first (that's just one data point, not the full pattern). Choices C and D misunderstand the concepts, while E incorrectly suggests the rule cannot be applied when it clearly can be used as an approximation.
Urban scholars describe the primate city pattern as one in which the largest city dominates political power, investment, and cultural influence, often shaped by colonial history or strong centralization. In Country Z, the capital contains most national ministries, top universities, and media; it has 9 million people, while the next largest city has 3.9 million. Which additional observation would most strongly support a primate city interpretation?
Small towns sell convenience goods, which is the defining feature of a primate city system.
The capital’s dominance demonstrates that primate city systems are always efficient and therefore always emerge.
The second-largest city is about half the size of the capital, which proves the country follows the rank-size rule.
Most major highways and rail lines radiate from the capital, reinforcing its dominance in flows of people and goods.
Because the primate city concept has limitations, evidence about transportation and ministries cannot be used to evaluate it.
Explanation
A primate city pattern involves the largest city dominating not just in population but in political, economic, and cultural functions. The capital already shows population dominance (9 million vs 3.9 million) and concentration of government, education, and media. Transportation infrastructure radiating from the capital (Choice A) would further reinforce its dominance by channeling flows of people, goods, and information through this hub. Choice B misapplies rank-size concepts. Choice C describes normal central place functions. Choice D wrongly claims primate patterns are always efficient. Choice E dismisses relevant evidence when transportation patterns clearly relate to urban primacy.
A secondary-source overview of urban systems argues that some countries show a primate city pattern, where the largest city is disproportionately large due to colonial administrative concentration, centralized investment, and dominance of national transportation and finance. In Country Y, the largest city has 12 million people, while the second-largest has 2.8 million and the third has 2.1 million. Which conclusion best matches the primate city concept?
Country Y reflects a hexagonal market-area system because the largest city is near the center of the country.
Country Y fits a primate city pattern because the largest city is far larger than the next cities, likely reflecting centralized political and economic power.
Country Y follows rank-size because the second city is smaller than the first, which is what the rank-size rule predicts.
Country Y cannot be described as primate because the concept ignores limitations like regional histories and state policies, so it should not be used.
Country Y proves primate cities are inevitable in all developing countries, since urban models describe reality rather than tendencies.
Explanation
A primate city pattern occurs when the largest city is disproportionately larger than other cities in the country, often due to centralized political and economic power. In Country Y, the largest city (12 million) is more than 4 times larger than the second city (2.8 million), showing clear dominance. This matches the primate city concept described in choice A. Choice B misunderstands rank-size rule, which would predict the second city to have 6 million people (12/2), not 2.8 million. Choice C incorrectly invokes hexagonal patterns from central place theory. Choice D overstates the model as inevitable rather than a tendency, while E incorrectly dismisses the concept entirely.
A secondary-source excerpt explains the primate city pattern: one city is disproportionately large and dominates economic, political, and cultural life, often linked to colonial administrative history and centralized investment. In Country Y, the capital has 12 million people, while the second-largest city has 3 million and most government ministries and corporate headquarters are in the capital. Which conclusion best aligns with the excerpt?
Because the capital is largest, the model guarantees the country has balanced regional development.
The primate city concept ignores all limitations, so it can be applied without considering history or regional inequality.
Country Y reflects the rank-size rule because the second-largest city is one-fourth the size of the largest.
Country Y’s pattern is best explained by central place theory’s assumption of evenly distributed consumers on a flat plain.
Country Y likely has a primate city, with dominance reinforced by centralized governance and historical investment.
Explanation
A primate city is one that is disproportionately large compared to others in the country, often dominating economic, political, and cultural aspects due to factors like centralized governance or colonial history. In Country Y, the capital with 12 million people is four times larger than the second-largest city of 3 million, and it hosts most government and corporate functions, fitting the primate pattern. This dominance suggests reinforced centralization, aligning with historical investments in the capital. Choice C accurately concludes this, emphasizing governance and investment as key reinforcers. In contrast, rank-size would predict the second city to be about half the largest, not one-fourth, so it does not apply here. The excerpt highlights how such patterns can lead to regional inequalities, not balanced development.
A secondary-source excerpt on deviations from urban models explains that rank-size and central place patterns often break down due to uneven terrain, colonial-era port concentration, resource booms, and state policies that favor a capital region. In Country Z, mountains isolate the east, a major port city grew rapidly during colonial trade, and the government heavily funds the capital’s infrastructure. Which claim best aligns with the excerpt?
Country Z’s uneven development could produce a primate city or regional imbalances that deviate from ideal hierarchies.
Country Z will necessarily have evenly spaced cities because mountains create hexagonal market areas.
Because the capital is funded, the country automatically becomes more regionally equal and less centralized.
Since models have limitations, they should never be used to interpret any city-size patterns.
Country Z must follow the rank-size rule because all countries do when they industrialize.
Explanation
Urban models like rank-size and central place theory provide ideal patterns, but real-world deviations occur due to factors such as terrain, colonial history, resource distribution, and government policies. In Country Z, mountains isolating regions, a colonial port's growth, and capital-focused funding could lead to uneven development and a primate city. This aligns with the excerpt by suggesting potential imbalances or deviations from smooth hierarchies. Choice B correctly claims this, highlighting how such factors produce primate patterns or regional disparities. Models remain useful for comparison, even when distorted, to analyze urban systems. Ignoring models entirely would overlook their interpretive value despite limitations.