Question

Suppose that D(r) is a function which gives the population density, in hundreds of people per square mile, within r miles of the center of a very small town. Assume that there are 100 people per square mile within 5 miles of the town’s center, and 350 people per square mile within 7 miles. a.Let T(r) be the number of people (in hundreds) who live within r miles of the city. Write an expression for T(r), in terms of r and D(r). b.What is the average rate of change of the function T (r) over the interval [5, 7]

Answer 1

Answer:

a) T(r) = r.D(r)

b) 975 people per square mile.

Step-by-step explanation:

(a)

The units of D(r) are  

$\frac{hundreds\;of\;people}{mile^2}$

when multiplying the density by the area we get the amount of people in that area, so

$T(r)=rmile^2D(r)\frac{hundreds\;of\;people}{mile^2}=rD(r)\;hundreds\;of\;people$

(b)

The average rate of change of T(r) over the interval [5,7] is

$\frac{T(7)-T(5)}{7-5}=\frac{7D(7)-5D(5)}{2}$

but D(7) = 350 and D(5) = 100, so

$\frac{7D(7)-5D(5)}{2}=\frac{7*350-5*100}{2}=975$

and the average rate of change of T(r) over the interval [5,7] is 975 people per square mile.