SECTION 1.6SECTION 1.6

MATHEMATICAL MODELS: MATHEMATICAL MODELS: CONSTRUCTING FUNCTIONSCONSTRUCTING FUNCTIONS

MAXIMIZING INCOMEMAXIMIZING INCOME

A car rental agency has 24 A car rental agency has 24 identical cars. The owner of identical cars. The owner of the agency finds that all the the agency finds that all the cars can be rented at a price cars can be rented at a price of $10 per day. However, for of $10 per day. However, for each $2 increase in rental, one each $2 increase in rental, one of the cars is not rented. of the cars is not rented. What should be charged to What should be charged to maximize income?maximize income?

DEMAND EQUATIONDEMAND EQUATION

In economics, revenue R is In economics, revenue R is defined as the amount of defined as the amount of money derived from the money derived from the sale of a product and is sale of a product and is equal to the unit selling equal to the unit selling price p of the product price p of the product times the number x of times the number x of units sold.units sold.R = xpR = xp

DEMAND EQUATIONDEMAND EQUATION

In economics, the Law of Demand In economics, the Law of Demand states that p and x are related: states that p and x are related: As one increases, the other As one increases, the other decreases.decreases.

Example: Suppose x and p Example: Suppose x and p obeyed the demand equation: x = obeyed the demand equation: x = - 20p + 500 where 0 - 20p + 500 where 0 << p p << 25. 25.

Express the revenue R as a Express the revenue R as a function of x.function of x.

DEMAND EQUATIONDEMAND EQUATION

x = - 20p + 500 where 0 x = - 20p + 500 where 0 << p p << 25.25.

Express the revenue R as a Express the revenue R as a function of x.function of x.

R = xp so in order to write R as R = xp so in order to write R as a function of x, we have to a function of x, we have to know what p is in terms of x know what p is in terms of x and then replace p with that and then replace p with that expression in R.expression in R.

DEMAND EQUATIONDEMAND EQUATION

x = - 20p + 500 where 0 x = - 20p + 500 where 0 << p p << 25.25.

x - 500 = - 20px - 500 = - 20p 25 x

201

- p

R = xpR = xp

25x x201

- R 2 Find the Find the maximum maximum Revenue.Revenue.

EXAMPLESEXAMPLES

DO EXAMPLES 3, 5, AND 6DO EXAMPLES 3, 5, AND 6

CONCLUSION OF SECTION 1.6CONCLUSION OF SECTION 1.6