Math 28 Project I
Introduction To start, consider the last time you purchased something, such as cereal. Let’s say you bought 5 boxes of cereal at $3.00. Now, what if there was a sale? You might be more likely to buy more than 5. However, if there was a price increase, you would probably buy less.
Price function For each price level of a product, there is a corresponding quantity of that product that consumers will demand (purchase) during some time period. Usually, the higher the price, the smaller is the quantity demanded; as the price falls, the quantity demanded increases. If the price per unit of the product is given by
pand the corresponding quantity is given by q, then an equation relating
pand
qis called a
demand equation. We’ll make a simplifying assumption that the demand equation is
linear. A good question for thinking deeper is how you address more complex types of equations, such as piecewise linear.
Since negative prices or quantities are not meaningful, both
pand
qare both nonnegative. If we solve for
pin the demand equation, the resulting equation is called a
price function. For most products, an increase in the quantity demanded corresponds to a decrease in price. Thus, a typical price curve
falls from left to right.
Below is an example of a price curve that goes through the points (0, 1000) and
(100, 800):
Figure 1 - Example price function. Price decreases as quantity sold increases.
In fact, we see that the price function is the line through the points (0, 1000) and (100, 800):
, which simplifies to .
Recall that
revenueis defined to be the income generated by the sale of goods or services:
So given our price function , the
revenue function R(q)is
Note that R is a quadratic function of q, with , b = 1000, and c = 0. Since a
maximumat the vertex :
.
Thus the maximum value of R is given by
This says that the maximum revenue that the manufacturer can make is $125,000, which occurs at a production level of 250 units.
Assignment:
Based on the above discussion, complete the problem below, showing your work in the spaces provided.
Through a series of marketing pilot programs, the marketing department determines the demand function for a small computer company’s premier laptop is , where
pis the price (in dollars) per unit when
qunits are demanded (per week) by consumers.
Find the level of production per week that will maximize total revenue, and determine the maximum revenue per week.
- Solve for
p
to determine the price function
p(q):
- Using the fact that Revenue = price x quantity, determine the revenue function R(q):
- Find the q-coordinate of the vertex, for the parabola that corresponds to R(q). This is the level of production per week that will maximize total revenue:
- Find the R-coordinate of the vertex, for the parabola that corresponds to R(q). This is the maximum revenue per week: