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
p and the corresponding quantity is given by q, then an equation relating
p and
q is 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
p and
q are both nonnegative. If we solve for
p in 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
revenue is 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
maximum at 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
p is the price (in dollars) per unit when
q units 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: