Chapter 9.4, Problem 47E

Solution

a.

Ten regions are used to categorise the states in the National Geographic Picture Atlas of Our Fifty States.

Heartland: $19,237,759$ people

Southeast: $42,614,977$ people

Given Information:

Determine Heartland's overall population:

  $\begin{array}{c}{P}_{\text{Heartland}}\text{}=2,926,324+2,688,418+4,919,479+5,595,211+1,711,283+642,200+754,844\\ =19,237,759\end{array}$

Determine Southeast's overall population:

  $\begin{array}{c}{P}_{\text{Southeast}}\text{}=4,447,100+2,673,400+15,982,378+8,186,453+4,468,976+2,844,658+4,012,012\\ =42,614,977\end{array}$

Conclusion:

Heartland: $19,237,759$ people

Southeast: $42,614,977$ people

b.

The total area of each region.

Heartland: $517,825$ square miles Southeast: $348,999$ square miles

Given Information:

Calculating Heartland's overall size

  $\begin{array}{c}{S}_{Heartland}\text{}=56,275+82,277+84,402+69,697+77,355+70,703+77,116\\ =517,825\end{array}$

Calculating Southeast's overall area:

  $\begin{array}{c}{S}_{Southeast}\text{}=51,705+53,187+58,644+58,910+47,751+47,689+31,113\\ =348,999\end{array}$

Conclusion:

Heartland: $517,825$ square miles Southeast: $348,999$ square miles

c.

The number of people per square mile in each region's population.

Heartland: $\approx 37.15$ people per square mile

Southeast: $\approx 122.11$ people per square mile

Given Information:

Count the number of people living in Heartland:

  $\begin{array}{c}{\rho }_{\text{Heartland}}=\frac{19,237,759}{517,825}\\ \text{}\approx 37.15\end{array}$ people per square mile

Determine Southeast's population density:

  $\begin{array}{c}{\rho }_{\text{Southeast}}\text{}=\frac{42,614,977}{348,999}\\ \text{}\approx 122.11 \end{array}$ people per square mile

Conclusion:

Heartland: $\approx 37.15$ people per square mile

Southeast: $\approx 122.11$ people per square mile

d.

The two areas, calculate each state's population density.

Given Information:

Determine the Heartland's population density in each state:

State	Population	Area	Density
Iowa	$2,926,324$	$56,275$	$52.00$
Kansas	$2,688,418$	$82,277$	$32.68$
Minnesota	$4,919,479$	$84,402$	$58.29$
Missouri	$5,595,211$	$69,697$	$80.28$
Nebraska	$1,711,283$	$77,355$	$22.12$
North Dakota	$642,200$	$70,703$	$9.08$
South Dakota	$7,54,844$	$77,116$	$9.79$

Identify the Heartland's typical density:

  $\frac{52+32.68+58.29+80.28+22.12+9.08+9.79}{7}\approx 37.75$

Identify the Southeast's states' densities of people:

State	Population	Area	Density
Alabarna	$4,779,736$	$51,705$	$86.01$
Arkansas	$2,915,918$	$53,187$	$50.26$
Florida	$18,801,310$	$58,644$	$272.53$
Georgia	$9,687,653$	$58,910$	$138.97$
Louisiana	$4,533,372$	$47,751$	$93.59$
Mississippi	$2,987,297$	$47,689$	$59.65$
South Carolina	$4,625,364$	$31,113$	$28.95$

Determine Southeast's typical density:

  $\frac{86.01+50.26+272.53+138.97+93.59+59.65+128.95}{7}\approx 118.57$ people per square mile

Conclusion:

The outcomes diverge from those that were partially obtained (c)

  $\rho =\frac{{\displaystyle {\sum }_{k=1}^7{P}_k}}{{\displaystyle {\sum }_{k=1}^7{S}_k}}$

While computing in portion (d)

  $\rho \prime =\frac{{\displaystyle {\sum }_{k=1}^7\frac{P_K}{S_k}}}{7}$