Question

Population Density Population density measures the number of people per square mile inhabiting a given living area. Washerton's population density, which decreases as you move away from the city center, can be approximated by the function \(10,000(2-r)\) at a distance \(r\) miles from the city center.(a) If the population density approaches zero at the edge of the city, what is the city's radius? (b) A thin ring around the center of the city has thickness \(\Delta r\) and radius \(r\) . If you straighten it out, it suggests a rectangular strip. Approximately what is its area? (c) Writing to Learn Explain why the population of the ring in part (b) is approximately $$10,000(2-r)(2 \pi r) \Delta r$$ (d) Estimate the total population of Washerton by setting up and evaluating a definite integral.

Short answer

a) The city's radius can be found as: R = 2 miles. b) The area of the thin ring can be approximated as: Area = \(2\pi r \Delta r\). c) The population is estimated by multiplying the area of the thin strip by the population density at radius \(r\): Population = \(10,000(2-r)(2\pi r) \Delta r\). d) The total population estimated using an integral results in: Population = \(\int_0^R 10,000(2-r)(2 \pi r) \Delta r\), where 'R' is the radius of the city.

Step-by-step solution

Determine the city's radius

Firstly, we know that the population density approaches zero at the edge of the city. This means we can set the formula for the population density, \(10,000(2-r)\), equal to zero, and solve for \(r\). Thus: \[10,000(2-r) = 0\] Solving this equation for \(r\) gives us the radius of the city.

Calculate the area of a thin ring

The area of a thin strip (ring) around the city can be approximated as the product of its circumference and its thickness \(\Delta r\). Given the circumference of a circle is \(2\pi r\), then the area can be expressed as \(2\pi r \Delta r\).

Explain the population of the ring

Now we want to find the population of the ring from part (b). The population is the product of the area of the strip and the population density at radius \(r\). Therefore, the population of the strip is approximately \(10,000(2-r)(2 \pi r) \Delta r\). This equation multiplies the population density \(10,000(2-r)\) by the area of the strip \(2\pi r \Delta r\) to give you the approximate population of the area.

Estimate the total population of Washerton

The last part of the question asks to estimate the total population by setting up and evaluating a definite integral. The integral is \(\int_0^R 10,000(2-r)(2 \pi r) \Delta r\), where R is the city's radius determined in step 1. The definite integral will sum up the approximate populations of all the infinitesimal thin rings (or strips) from the city center (r = 0) all the way to the city limits (r = R), thus giving the total population.