1. Section 9.3 Problem Solving Involving Max and Min
i) Revenue problems
ii) Maximum area
iii) Minimum for sums of two squares
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2. I) Basics in Operating a BUSINESS!
When we increase the price of an merchandise,  Sales will decrease
In contrast, when we decrease the price  sales will increase
Revenue is the income generated from sales
In this lesson, we will learn to Maximize Revenue and Profit
Ex#1) An ice-cream shop sells 300 cones a day at $3.50 each. For every $0.50 increase, he loses 20 sales.
Q:How does the price affect quantity?
Q:How can the price be changed to generate the maximum revenue?
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3. Analyze the question: Price vs Quantity
Initial Quantity:
Initial Price:
Change in Quantity:
Change in Price:
Sales begin at 300 cones per day
The initial price is at 3.50 per cone
Decrease of 20 cones in sales
Price is increased by $0.50/cone
3.5 4
4.5 5
5.5
200
300
Initial value
The points can be used to make a straight line
For every increase in 50 cents:
Use the slope equation to find how the price affects quantity
Sales will decrease by 20 units
For every change, a new set of coordinates will be created
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4. a) How does the price affect quantity?
The x-axis corresponds with the price
The y-axis corresponds with the quantity
If we plug in the values and Isolate “Q”, we obtain an equation that shows how the price affects quantity
This equation shows how the price affects the quantity
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5. b) What price will generate Maximum Revenue?
The points come together and generate a parabola
Revenue = Quantity x Price 1 2 3 4 5 6 7 8 9
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
The maximum revenue will be at the vertex (highest point on the graph)
Maximum revenue occurs when the price is at $5.50
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6. Finding the Maximum Revenue
We can find the maximum revenue by CTS or XAV
The vertex of the revenue equation indicates the maximum revenue: x-coord: Price , y-coord: Revenue
CTS with the revenue equation
The Maximum revenue occurs when the price is at $5.50
The Max revenue is $1210
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7. Write “Q” as a function of “p” Write “P” as a function of “Q”
Ex#2) A Broadway Musical sells 3500 tickets a week at $25 per ticket. For every $1.25 decrease, 400 extra tickets will be sold.
Write “Q” as a function of “p”
Write “P” as a function of “Q”
Write “R” as a function of “p”
What price will generate the maximum revenue?
What Price will generate a Revenue greater than $80,000
Use this chart to generate the equation:
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8. Write “P” as a function of “Q” (Isolate “P”)
Write “P” as a function of “R” and find the maximum revenue
The Maximum Revenue is $ and when the unit price is $17.97
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9. What price will yield a revenue greater than $80,000?x y -2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
2x10^4
4x10^4
6x10^4
8x10^4
10^5
1.2x10^5
Find the Intersection Point
For any price between $9.43 to $26.51, the revenue will be greater than $80,000!!
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10. To solve for the intersection points, make the Revenue equal to $80,000 and solve for the price
Isolate “P”
The Revenue will be greater than $80,000 if the price is between $9.43 to $26.51
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11. Homework: Assignment 9.3b
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