f ‘(x) = -2x + 200 = 0

Hence: x = 100

c. Graph the function:

Profit

The given graph is parabolic.

As expected, the profit is positive from x = 20 to x =180 verifying our answer in b. It

can also be seen that the maximum profit can be found in x = 100.

2. The population of the country is 50 million. Two months ago, the government required its citizen to purchase an identity card. After one month, 6 million people had it and by the end of the second month, 10 million people had one.

a. Model N (number of cards) as a function of t (months) using the form:

N = a + b/ (t+c)

Given are the following:

When t = 1 then N = 6,000,000 hence: 6,000,000 = a + b/(1+c) [eq.1]

t = 2 N = 10,000,000 hence: 10,000,000 = a + b/(2+c) [eq.2]

and of course, when

t = 0 N = 0 hence: 0 = a + b/c or a= -b/c [eq.3]

Simplifying eq. 1 and inserting the value of a from eq. 3, we have:

6,000,000 = -b/c + b/(1+c)

6,000,000 = (-b -bc +bc) / [c*(1+c)]

6,000,000 = -b/[c*(1+c)] but -b/c = a

6,000,000 = a / (1+c)

a = 6,000,000 + 6,000,000c [eq.4]

Simplifying eq. 1 and inserting the value of a from eq. 3, we have:

10,000,000 = -b/c + b/ (2+c)

10,000,000 = (-2b -bc +bc) / [c*(2+c)]

10,000,000 = -2b/[c*(2+c)] but -b/c = a

10,000,000 = 2a / (2+c)

a = 10,000,000 + 5,000,000c [eq.5]

By equation 4 &. 5, we can get easily get the value of a &. c:

a = 30,000,000

c =4

By eq. 3, a = – b/c or b = -ac. Hence: b = -120,000,000

The model equation is therefore: N = 30,000,000 + (-120,000,000)/ (t +4)

b. What is the function called Graph and define its features:

The function is of the type Rational function because the equations can be

expressed as a ratio of two polynomial…

The intersections of the revenue and cost lines represent also the value where there is no profit or loss. In addition, it can be seen that costs and average costs intersect at some point. This can be determined with the following solution:

6. Market research suggests that potential market for a product is 800,000. At year 1, the market penetration has reached 50% or 400,000. At year 2, the market penetration has reached 75% or 600,000. Using the following model below, answer the following questions: