Practice Problems 3.1
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Modeling a Pizza Party
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Today, we'll explore how to model a scenario using mathematics. Letβs say you're planning a pizza party for 20 people, and each person is expected to eat 3 slices. How do we start?
I think we first need to figure out how many total slices of pizza we need!
Exactly! If each person eats 3 slices, how do we calculate that for 20 people?
We can multiply 20 by 3, so that's 60 slices.
Correct! Now, since each pizza has 8 slices, how many pizzas do we need to order?
I guess we divide 60 by 8.
Right! Can someone do that calculation for me?
That equals 7.5! So we would need to order 8 pizzas, right?
Yes! Remember, you have to round up because you can't order half a pizza. If each pizza costs $15, how do we find the total cost?
We multiply 8 by 15, which is $120.
Great! So we have 8 pizzas at a total cost of $120. Let's summarize: We calculated the total slices needed, the number of pizzas, and the total cost.
Modeling a Phone Plan
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Now, let's transition to our second problem regarding a phone plan that costs $25 a month, and there's an additional charge of $0.10 for every minute over 100 minutes. How can we create a mathematical model for this?
We should define our variables first. Let's let 'm' be the total minutes used.
Exactly! So what is the expression for the total cost 'C' if the usage exceeds 100 minutes?
If m is greater than 100, the cost C = $25 + $0.10(m - 100).
Well done! Now, what would be the total cost if someone used 150 minutes?
The cost would be C = 25 + 0.10 Γ (150 - 100) = 25 + 0.10 Γ 50 = 25 + 5 = $30.
Perfect! Now, how would we find out how many minutes that user used if their bill was $30?
We would set the equation $30 = $25 + $0.10(m - 100) and solve for m.
Exactly! What would that be?
That gives us $30 - $25 = $0.10(m - 100), so $5 = $0.10(m - 100), leading to m = 150 minutes.
Excellent work, everyone! Today, we learned how to model real-world problems using mathematical equations and found our solutions using clear steps.
Introduction & Overview
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Quick Overview
Standard
This section contains two practical problems that encourage students to apply mathematics to real-world situations. Each problem involves modeling and requires students to utilize their knowledge of mathematical concepts to find solutions regarding a pizza party and a phone plan.
Detailed
Practice Problems 3.1
In this section, we focus on two engaging scenarios that require mathematical modeling, encouraging students to apply their acquired skills and knowledge from previous units. The first problem revolves around planning a pizza party for 20 people, where each guest is expected to consume 3 slices. Students are tasked to determine the number of large pizzas needed and the total cost associated with the order. This exercise tests their understanding of basic multiplication, division, and budgeting.
The second problem presents a phone plan that charges a basic monthly fee plus an additional cost per minute for calls exceeding a set limit. Students will create a mathematical model to express the total cost based on their usage. This problem emphasizes the importance of formulating equations based on real-world conditions, thereby bridging the gap between abstract math and its practical application.
Both problems serve to reinforce the concept of mathematical modeling and problem-solving, as students must translate a real-world context into mathematical terms and subsequently justify their reasoning.
Audio Book
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The Pizza Party Problem
Chapter 1 of 2
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Chapter Content
- Problem: Model this scenario to determine how many pizzas you need to order and the total cost.
- Hint: What are your variables? What assumptions are you making (e.g., everyone eats exactly 3 slices)?
Detailed Explanation
In the pizza party problem, we need to calculate how many pizzas to order based on how many people are attending and how many slices each person will eat. First, we need to find out the total number of slices needed. Since there are 20 people and each expects to eat 3 slices, we multiply these two numbers: 20 x 3 = 60 slices. Next, we find out how many pizzas we need to order. Each pizza has 8 slices. Therefore, we divide the total slices by the number of slices per pizza: 60 Γ· 8 = 7.5. Since you can't order half a pizza, you need to round up to 8 pizzas. Finally, we calculate the total cost. If each pizza costs $15, then the total cost is 8 x $15 = $120.
Examples & Analogies
Imagine you are hosting a birthday party for your friend. You invited 20 friends and each friend, being a pizza lover, says they'll eat 3 slices. You first count how many slices you'll need in total, just like a chef estimating how much pizza to prepare. After figuring out how many whole pizzas you need, you place your order, ensuring everyone at the party leaves satisfied and full!
The Phone Plan Problem
Chapter 2 of 2
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Chapter Content
- Problem: Create a mathematical model (an equation) to find the total monthly cost (C) based on the number of minutes (m) used, assuming m is greater than 100.
- Then, use your model to find the cost if you use 150 minutes in a month.
- And, find how many minutes you used if your bill was $30.
Detailed Explanation
In this phone plan problem, the cost consists of a base fee plus an additional charge for minutes over 100. The base fee is $25, and each extra minute costs $0.10. To model this, we can create an equation: C = 25 + 0.10(m - 100). This equation allows us to calculate the total monthly cost based on the number of minutes used (m) beyond the first 100 minutes. For example, if m = 150 minutes, we substitute into the equation: C = 25 + 0.10(150 - 100) = 25 + 0.10(50) = 25 + 5 = $30. To find out how many minutes used corresponds to a bill of $30, we set C to 30, leading us to: 30 = 25 + 0.10(m - 100). Subtracting 25 from both sides gives us 5 = 0.10(m - 100). Dividing by 0.10 gives us m - 100 = 50; therefore, m = 150.
Examples & Analogies
Think of your phone plan like a gym membership. You pay a flat fee to be able to enter the gym but get charged extra if you use certain equipment beyond a basic amount. Just like when you estimate how much you might spend in the gym based on how many extra classes or personal training sessions you'll attend, in this phone plan example, you're calculating how much your phone bill will be based on your usage beyond the basic plan!
Key Concepts
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Mathematical Modeling: The practice of representing real-world problems with mathematical expressions and equations.
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Cost Calculation: Understanding how to determine total expenses in different scenarios.
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Variables and Equations: The importance of defining variables and forming equations in problem-solving.
Examples & Applications
Example for Pizza Party: Calculating total slices needed for 20 people who eat 3 slices each leads us to need 8 pizzas.
Example for Phone Plan: For a bill of $30 with the provided phone plan, calculating how much you would spend helps determine usage.
Memory Aids
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Rhymes
For pizza, plan your party with flair, 3 slices per guest, show that you care!
Stories
Imagine youβre throwing a grand pizza party, where everyone's invited, and you need to count the pies like a puzzle, so no one's left hungry or unsatisfied!
Memory Tools
To remember how to calculate party needs, think P.E.T. (People x Edibles x Total Cost)!
Acronyms
C.P.M. - Cost, Pizza, Minutes for planning your events efficiently.
Flash Cards
Glossary
- Modeling
The process of representing a real-world situation using mathematical equations or expressions.
- Variable
A symbol (such as x or m) used to represent a number that can change or vary.
- Equation
A mathematical statement that asserts the equality of two expressions.
- Total Cost
The overall amount that has to be paid, calculated by considering all expenses involved in the context.
- Assumptions
Conditions that are accepted as true or as certain to happen, used to simplify problems.
Reference links
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