Pack-Em-In Real Estate is building a new housing development. The more houses it builds, the less people will be willing to pay, due to the crowding and smaller lot sizes. In fact, if it builds 40 houses in this particular development, it can sell them for $200,000 each, but if it builds 60 houses, it will only be able to get $160,000 each. Obtain a linear demand equation and hence determine how many houses Pack- Em-In should build to get the largest revenue. What is the largest possible revenue?

Using a linear and a quadratic function, it is found that the largest possible revenue is of $9,800,000.

The demand function has a linear format, given by:

In which:

40 houses can be sold for $200,000 each, thus, the first point of the linear function is: (40, 200000).

60 houses can be sold for $160,000 each, thus, the second point is: (60, 160000).

The slope is given by change in the output divided by change in the input, thus:

Thus:

Point (40, 200000) means that when . This is used to find b.

Then

The revenue function is:

It is a quadratic equation with

Since the quadratic equation is concave down, the maximum value is:

The largest possible revenue is of $9,800,000.

A similar problem is given at link

so the possibilities of the measurement of the side of the triangle are either it is longest side or not the longest side

when it is the longest side

s < 25 + 35s < 60

if it is not the longest side

35 < 25 + s10 < sso 10 < s < 60so the perimeter70 < p < 130