Question

explain 1b using area
visit bim.easyaccessmaterials.com, read integrated mathematics 3 les
teacher voice - you can use the formulas for population density to find missing information about the
for example, you can find the radius of a circular region given the population and the population density
example 2 - using the formula for population density
a circular region has a population of about 175,000 people and a population density of about 1318 people per
square mile. find the radius of the region.
solution
population density = \\(\frac{number\\ of\\ people}{area\\ of\\ land}\\)
population density formula
\\(1318 = \frac{175,000}{\pi r^2}\\)
substitute the given values and the
formula for the area of a circle.
\\(1318 r^2 = \frac{175,000}{\pi}\\)
multiply each side by \\(\pi r^2\\).
\\(r^2 = \frac{175,000}{1318 \pi}\\)
divide each side by 1318.
\\(r = \sqrt{\frac{175,000}{1318 \pi}}\\)
take the square root of each side.
\\(r \approx 6.5\\)
use a calculator.
the radius of the region is about 6.5 miles.
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try this video for extra example 2 - using the formula for population density

1. a circular region has a population of about 21,400 people and a population density of about 425

people per square mile. find the radius of the region.

Explanation:

Step1: Recall population density formula

The formula for population density is \( \text{Population density} = \frac{\text{number of people}}{\text{area of land}} \). For a circular region, the area of land is \( \pi r^2 \), where \( r \) is the radius. So we substitute into the formula: \( 425=\frac{21400}{\pi r^{2}} \).

Step2: Multiply both sides by \( \pi r^2 \)

To eliminate the denominator, we multiply each side of the equation \( 425=\frac{21400}{\pi r^{2}} \) by \( \pi r^2 \). This gives us \( 425\pi r^{2}=21400 \).

Step3: Divide both sides by \( 425\pi \)

Now, we want to solve for \( r^2 \), so we divide each side of the equation \( 425\pi r^{2}=21400 \) by \( 425\pi \). So \( r^{2}=\frac{21400}{425\pi} \).

Step4: Take the square root of both sides

To find \( r \), we take the square root of both sides of the equation \( r^{2}=\frac{21400}{425\pi} \). This gives \( r = \sqrt{\frac{21400}{425\pi}} \).

Step5: Calculate the value

First, simplify the fraction inside the square root: \( \frac{21400}{425}=\frac{21400\div25}{425\div25}=\frac{856}{17}\approx50.35 \). Then, \( \pi\approx3.14 \), so \( 50.35\div(3.14)\approx16.03 \). Then take the square root of \( 16.03 \), so \( r\approx\sqrt{16.03}\approx4 \) (we can also calculate directly using a calculator: \( \sqrt{\frac{21400}{425\times3.14}}=\sqrt{\frac{21400}{1334.5}}\approx\sqrt{15.99}\approx4 \)).

Answer:

The radius of the region is about \( 4 \) miles.