ANSWER ALL PLEASE

Project $4-$ MHF$4$U

If you build it$,$ will they come?

Suppose that you are a financial advisor for a building contractor. Your client has been offer a

contract to build up to $500$ new houses but is not sure how many she should build. If she builds

to fee, she might not make enough money to cover the expenses. If she builds too many, market

saturation might drive down prices, resulting in a risk of a loss. Is there an optimum number that

you would advise your client to build?

Often in business, as well as in other disciplines, analysing combine functions can lead to a range

of acceptable solutions. In such cases, techniques for solving inequalities are often useful.

Deeper analysis can also reveal conditions for an optimum solution.

Task:

A building developer can build up to $500$ houses for a new subdivision for the following costs:

Fixed cost: $$8000000$

Variable cost: $$65000$ per house

The developer earns a revenue, $R,$ in millions of dollars, from sales according to the function,

$R(n)=1\mathrm{.}6\sqrt[\phantom{2}]{n},$ where $n$ is the number of houses.

Write an equation to represent the total cost, $C,$ in millions of dollars, as a function of the

number, $n,$ of houses built and sold.

Graph $R(n)$ and $C(n)$ on the same set of axes. Choose appropriate window settings so

that sales of between $0$ and $500$ houses can be viewed.

Identify the region where $R(n)>C(n).$ Sketch the graph and shade in this region. What

is its significance?

Identify the points of intersection of the two curves and determine their coordinates using

a method of your choice. Explain the meaning of this coordinates.

A$)$ Use the superposition principle to graph $y=R(n)-C(n).$

B$)$ Describe at least three interesting things that this graph shows.

Reflection:

Write a brief letter to the housing developer advising about the subdivision contract

which includes:

Minimum and maximum number of houses that she should agree to build.

The number of houses she would need to build and sell to maximum profit

Suggestions of how she might able to increase the profit. Use mathematical

thinking to support your advice.