Alg1 Lesson 3.3: Revenue and Profit

Lesson 3.3: Revenue and

Profit

Obj: 6A, 6B, 6C, 7A, 8B: I can interpret the vertex
and roots of a quadratic model in context.

EQ: How do I formulate equations given a context?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

Word Cloud

How do you manage your time?

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Part 1: Warming Up with

Economic Principles

Lemonade Stand

Miles and Lucy decide to sell lemonade to earn some money
to go to the movies. They buy supplies such as pitchers, ice,
and lemonade mix for a cost of $13.50. The lemonade mix
makes 8 quarts of lemonade. They set up shop, and decide to
sell one cup of lemonade for a price of $0.75. If they sell all 8
quarts of lemonade, how much revenue will they make, and
will they make enough profit to buy movie tickets for $12
each? (There are 4 cups in 1 quart.)

Drag and Drop

Term & meaning from the lemonade stand problem.

Cost: ​

Placeholder

Price: ​

Placeholder

Revenue: ​

Placeholder

before deducting expenses.
Profit: ​

Placeholder

Drag these tiles and drop them in the correct blank above

the amount of money you spend on expenses.

the amount you will charge

the amount of $ you generate from selling

is the amount of money you have left after sales.

Drag and Drop

Term & $ from the lemonade stand problem.

Cost: ​​

Placeholder

Price: ​

Placeholder

Revenue: ​

Placeholder

Profit: ​

Placeholder

Drag these tiles and drop them in the correct blank above

$13.50

$0.75

$24

$10.50

$8

$12

Profit = Revenue - Cost = Price(items sold) - Cost

Part 2: Finding the Best

Price

Pie-Selling Scenario

Every year, the math club at your school sells pies on Pi Day as
a fundraiser. Since you have done this for several years, you
have data to use in setting the price. When you made 160 pies,
you sold them all at $6 per pie. When you made 80 pies, you
sold them all for $10 each. Your principal has allowed you to use
the school cafeteria kitchen for a one-time, nonrefundable fee of
$300, and the ingredients for each pie cost $2. How many pies
should you make to maximize profit? How much should you sell
the pies for? How much money will you make?

Questions to Consider

If the price of the pies is really high, do you think we will sell a lot of pies or not that many pies? Why?


If we know we’re not going to sell a lot of pies, will our costs be high or low? Why?


If the price of the pies is really low, do you think we will sell a lot of pies or not that many pies? Why?


If we know we are going to sell a lot of pies, will our costs be high or low? Why?


If the price is high, we don’t sell that many pies, but the costs are low. If the price is low, we sell a lot of pies but the costs are high. What might happen to our revenue and profit in each case?

The Optimization Challenge

Finding a pie price that will make the costs
low enough and the revenue high enough to
make a profit.

Word Cloud

What comes to mind when you think about the word, "Profit"?

Formulate Functions for each output variable. Multivariate

Building the Cost Function C(x)

What do we know about the costs for making pies?

What do these quantities mean?

What would be a linear function that models the cost in
terms of the number of pies?

Drag and Drop

The use of the kitchen is going to have an initial cost of ​

Placeholder

and the ingredients for each pie will be ​

Placeholder

.
A linear function that models the cost in terms of the number of pies is

C(x) = ​

Placeholder

x + ​

Placeholder

Drag these tiles and drop them in the correct blank above

$300

$2

$6

6

80

2

300

Building the Price Function E(x)

What information are we given about the price of the pies?

In general, does it look like the number of pies sold
increases or decreases as the price get higher?

How can we represent the information as two ordered
pairs of numbers?

What would be a linear function that models the price in
terms of the number of pies?

Drag and Drop

When you made ​

Placeholder

pies, you sold all the pies at $6 per pie.
When you made ​

Placeholder

pies, you sold all the pies for

$​

Placeholder

each.
The input is the number of pies, and the output is the price of each pie.

Using the points (​ 160, 6) and (​

Placeholder

, ​

Placeholder

) we can write the linear function for the price.

Drag these tiles and drop them in the correct blank above

6

80

160

10

2

300

Multiple Choice

What would be a linear function that models the price in terms of the number of pies?

• E(x) = 3x - 5
• E(x) = -1/20x + 14
• E(x) = 2x + 10
• E(x) = -1/10x + 20
• E(x) = -1/5x + 12

Building the Revenue Function R(x)

If we sold 15 pies for $6 each, the revenue would be $90
since 15*6=90

If we sold 1 pie for $6, the revenue would be $6 since 1*6=6

If we sold 378 pies for $6 each, the revenue would be $2,268
since 378*6=2268

What would be a quadratic function that models the revenue?

Recall that revenue = items sold times price

Multiple Choice

How do you calculate revenue if you know the price per item
and the number of items you sell?

• Divide the price per item by the number of items sold

• Add the price to the number of items.

• Multiply the price by the
  number of items.

• Subtract the price from the number of items.

Multiple Choice

What would be a function that models the revenue in terms of the number of pies and the price per pie?

• R(x) = x(10 - 1/10x)
• R(x) = x(16 - 1/25x)
• R(x) = x(14 - 1/20x)
• R(x) = x(12 - 1/15x)
• R(x) = x(20 - 1/5x)

Graphing Revenue Function

What will the revenue be if we sell 10 pies?

If we don’t sell any pies, what will our revenue be?

Which of the quantities should we display on the horizontal and vertical axes?

What quadrants will be used for the graph of the revenue?

Could the number of pies (the input) be a decimal or fraction? Could the revenue
generated (the output) be a decimal or fraction?

Looking at the function, would you predict that the graph will be a line or another
shape? What is your justification? What might be the shape of the graph?

Fill in the Blank

What will the revenue be if we sell 10 pies?

$

Fill in the Blank

If we don’t sell any pies, what will our revenue be?

$

Drag and Drop

The horizontal axis should display the input, so ​

Placeholder

. The vertical axis should display the output, ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

the number of pies sold

so the revenue generated by selling pies

Multiple Select

What quadrant(s) will be used for the graph of the revenue?

• I

• II

• III

• IV

Multiple Choice

Could the number of pies (the input) be a decimal or fraction? Could the revenue generated (the output) be a decimal or fraction?

• yes, yes

• yes, no

• no, yes

• no, no

Multiple Choice

What type of function would the revenue function be modeled by?

• Linear

• Quadratic

• Exponential

• Rational

Drag and Drop

What is the shape of this graph? ​

Placeholder

Determine the coordinates of the vertex.

The vertex is (​

Placeholder

, ​

Placeholder

).

What does that ordered pair of numbers mean in the context of the problem?

That ​

Placeholder

Is our goal to maximize revenue? ​

Placeholder

Drag these tiles and drop them in the correct blank above

Parabola

980

140

No

Yes

140 pies will give the most revenue: $980.

​980 pies will give the most revenue: $140.

Building the Profit Function P(x)

The profit is P(x) = R(x) - C(x) = x(14-(1/20)x)-(2x+300)

Which part of the graph will reveal the maximum profit?

What are the coordinates of this point?

What does each number represent in the context of
the problem?

How many pies to make to maximize profit?
120 pies

How much should you sell the pies for? $___

How much money will you make (if you sold all pies)?

$420

Multiple Choice

How much should you sell the pies for?

• $8

• $6

• $10

• $12

• $2

Optimization

Optimization often involves making a tradeoff
between competing factors or constraints.
Collecting data about how many people buy a
product at what price can help determine what
sale price is best and finding the maximum profit
can determine how many items to manufacture.

Multiple Choice

The profit is P(x) = R(x) - C(x) = x(14-(1/20)x)-(2x+300). What is this in standard form?

• P(x) = -1/20x^2 + 12x - 300
• P(x) = 1/20x^2 + 12x + 300
• P(x) = -1/20x^2 - 16x + 300

• P(x) = 1/20x^2 - 16x - 300

Drag and Drop

What does the y-intercept represent for the pie business scenario?
Before we make any pies, there is a $​

Placeholder

nonrefundable ​

Placeholder

to use the kitchen.

Placeholder

money. The additional ​

Placeholder

exceeds the additional ​

Placeholder

we would generate.

Drag these tiles and drop them in the correct blank above

300

cost

lose

gain

revenue

price

profit

420

Multiple Choice

What is the smallest number of pies we need to sell to break even?

• 28

• 0

• 120

• 250

• 211

Multiple Choice

What is the largest number of pies we can sell and still make a profit, even if it’s not the maximum profit?

• 28

• 500

• 120

• 250

• 211

Poll

Do you feel like you know more about finances after these slides?

• Yes

• No

Part 3: Defining Quadratic

Functions

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin the
wheel as a class and spend a minute or so
discussing our answers.