Question

Sun Valley has 15,000 people living in an area of 12.5 square miles.Pearce City has 22,500 people living in an area of 15 square miles. Which city is likely morecrowded?

Answer 1

We will pluck out the data from the given problem. We will express the number of people ( population) in two cities ( P1 and P2 ) and the area over which the people are spread in each city as ( A1 and A2 ).

Sun Valley:

`\begin{gathered} Population(P_1\text{) = 15,000 people } \\ \text{Area of spread ( A}_{1\text{ }})=12.5miles^2 \end{gathered}`

Peatce City:

`\begin{gathered} \text{Population ( P}_2\text{ ) = 22,500 people} \\ \text{Area of spread ( A}_2)=15miles^2 \end{gathered}`

We are to compare the two cities on the basis of "crowding". We will first interpret what is meant by crowding in the context of this question.

Crowding is termed as number of people in a unit area. This means a city with more number of people in a small area would be more crowded than a city with less number of people in large area.

Mathematically speaking, its the ratio of Population ( P ) in each city to the area coverage ( A ) over which the population is spreaded to. Statisticians represents this ratio as " Population Density ( PD )".

It is a useful parameter used in various statistical analysis at higher proects level.

We will compute the population density of each city as follows.

SunValley:

`\begin{gathered} PD_1\text{ = }(P_1)/(A_1)\text{ = }(15,000)/(12.5) \\ \textcolor{#FF7968}{PD}_{\textcolor{#FF7968}{1}}\text{\textcolor{#FF7968}{ = 1,200 }}\textcolor{#FF7968}{(people)/(miles^2)} \end{gathered}`

Pearce City:

`\begin{gathered} PD_2\text{ = }(P_2)/(A_2)\text{ = }(22,500)/(15) \\ \textcolor{#FF7968}{PD_2}\text{\textcolor{#FF7968}{ = 1,500}}\textcolor{#FF7968}{(people)/(miles^2)} \end{gathered}`

Note: The reason why we used population density as a paramter for crowding is because the population and coverage area are both different in each city. So for adequate comparison we equalize one of the two quantities ( Population or Area ) for both cities. In the parameter of " population density - PD " we equated the coverage area. We can see this from the units of poluation density ( PD1 and PD2 ) where the coverage area is equated to 1 square mile.

Now, once we have the required parameter for "crowding". We can compare the population densities of each city as follows:

`\begin{gathered} PD_2>PD_1 \\ 1,500\text{ > 1,200} \end{gathered}`

Therefore, we see that Pearce City has more number of people residing in a square mile than the number of people in a square mile at Sun Valley! Hence,

The city which is most likely to be crowded is the one with higher population density:

`\text{\textcolor{#FF7968}{Pearce City}}`