Question

find the indicated measure. (see example 1)

1. the state of kansas has a population of about 2.85 million people. find the population density in people per square mile.

error analysis

1. describe and correct the error in finding the number of people who live in a circular region with a 7.5-mile diameter and a population density of 1550 people per square mile. (see example 1)

problem solving (see example 2

1. about 79,000 people live in a circular region with a population density of about 513 people per square mile. find the radius of the region. (see example 2)

Explanation:

Problem 3:

Step 1: Find the area of Kansas

Kansas is approximately a rectangle, so the area \( A \) is length times width. The length is 400 miles and the width is 210 miles. So, \( A = 400 \times 210 \) square miles.
\( A = 84000 \) square miles.

Step 2: Convert population to number

The population is 2.85 million, which is \( 2.85 \times 10^6 = 2850000 \) people.

Step 3: Calculate population density

Population density \( d \) is given by the formula \( d=\frac{\text{Population}}{\text{Area}} \). So, \( d = \frac{2850000}{84000} \).
Simplifying, \( d\approx33.93 \) people per square mile.

The error is in calculating the radius. The diameter is 7.5 miles, so the radius \( r=\frac{7.5}{2}=3.75 \) miles, not 7.5 miles. The correct formula for the area of a circle is \( A = \pi r^2=\pi(3.75)^2=\pi\times14.0625 \). Then, using population density \( d=\frac{\text{Population}(x)}{\text{Area}(A)} \), so \( x = d\times A \). Substituting \( d = 1550 \) and \( A=\pi\times14.0625 \), we get \( x = 1550\times\pi\times14.0625\approx1550\times44.1786\approx68477 \).

Step 1: Recall the formula for population density

Population density \( d=\frac{\text{Population}(P)}{\text{Area}(A)} \), and for a circle, \( A=\pi r^2 \). So, \( d=\frac{P}{\pi r^2} \), and we can solve for \( r \).

Step 2: Rearrange the formula for \( r \)

From \( d=\frac{P}{\pi r^2} \), we get \( r^2=\frac{P}{d\pi} \), then \( r = \sqrt{\frac{P}{d\pi}} \).

Step 3: Substitute the values

Given \( P = 79000 \) people and \( d = 513 \) people per square mile. So, \( r=\sqrt{\frac{79000}{513\times\pi}} \). First, calculate the denominator: \( 513\times\pi\approx513\times3.1416\approx1611.64 \). Then, \( \frac{79000}{1611.64}\approx49.02 \). Then, \( r=\sqrt{49.02}\approx7.00 \) miles.

Answer:

The population density of Kansas is approximately \( \boldsymbol{33.93} \) people per square mile.

Problem 11: