A Movie Theater Is Giving Away A Souvenir Poster To Any Customer With A Concession Stand Receipt That Exceeds $ 60 \$60 $60 . The Theater Sells A Bag Of Popcorn For $ 6 \$6 $6 And A Bottle Of Soda For $ 3.50 \$3.50 $3.50 . Let X X X Represent

Introduction

A movie theater is giving away a souvenir poster to any customer with a concession stand receipt that exceeds $\$60$. The theater sells a bag of popcorn for $\$6$ and a bottle of soda for $\$3.50$. Let $x$ represent the number of bags of popcorn and $y$ represent the number of bottles of soda. In this article, we will explore the conditions under which a customer can receive the souvenir poster and analyze the mathematical relationships between the number of popcorn bags, soda bottles, and the total cost.

The Problem

To receive the souvenir poster, a customer's concession stand receipt must exceed $\$60$. This means that the total cost of the popcorn bags and soda bottles must be greater than $\$60$. Mathematically, this can be represented as:

$$6x + 3.50y > 60$$

where $x$ is the number of bags of popcorn and $y$ is the number of bottles of soda.

Analyzing the Inequality

To analyze the inequality, we can start by isolating the variable $y$. We can do this by subtracting $6x$ from both sides of the inequality:

$$3.50y > 60 - 6x$$

Next, we can divide both sides of the inequality by $3.50$ to get:

$$y > \frac{60 - 6x}{3.50}$$

Understanding the Relationship Between x and y

The inequality $y > \frac{60 - 6x}{3.50}$ shows that the number of bottles of soda ($y$) must be greater than a certain value that depends on the number of bags of popcorn ($x$). This means that as the number of bags of popcorn increases, the number of bottles of soda required to exceed $\$60$ also increases.

Graphing the Inequality

To visualize the relationship between $x$ and $y$, we can graph the inequality on a coordinate plane. The graph will consist of a region above the line $y = \frac{60 - 6x}{3.50}$, where the number of bottles of soda is greater than the required value.

Finding the Minimum Number of Bags of Popcorn

To find the minimum number of bags of popcorn required to exceed $\$60$, we can set $y = 0$ and solve for $x$:

$$6x > 60$$

Dividing both sides of the inequality by $6$ gives:

$$x > 10$$

This means that the customer must buy at least $11$ bags of popcorn to exceed $\$60$.

Finding the Minimum Number of Bottles of Soda

To find the minimum number of bottles of soda required to exceed $\$60$, we can set $x = 0$ and solve for $y$:

$$3.50y > 60$$

Dividing both sides of the inequality by $3.50$ gives:

$$y > 17.14$$

This means that the customer must buy at least $18$ bottles of soda to exceed $\$60$.

Conclusion

In conclusion, a customer can receive the souvenir poster if their concession stand receipt exceeds $\$60$. The total cost of the popcorn bags and soda bottles must be greater than $\$60$, and the number of bottles of soda must be greater than a certain value that depends on the number of bags of popcorn. By analyzing the inequality and graphing the relationship between $x$ and $y$, we can find the minimum number of bags of popcorn and bottles of soda required to exceed $\$60$.

Introduction

In our previous article, we explored the conditions under which a customer can receive a souvenir poster from a movie theater's concession stand. We analyzed the mathematical relationships between the number of popcorn bags, soda bottles, and the total cost. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the minimum number of bags of popcorn required to exceed $60?

A: To find the minimum number of bags of popcorn required to exceed $60, we can set y = 0 and solve for x:

6x > 60

Dividing both sides of the inequality by 6 gives:

x > 10

This means that the customer must buy at least 11 bags of popcorn to exceed $60.

Q: What is the minimum number of bottles of soda required to exceed $60?

A: To find the minimum number of bottles of soda required to exceed $60, we can set x = 0 and solve for y:

3.50y > 60

Dividing both sides of the inequality by 3.50 gives:

y > 17.14

This means that the customer must buy at least 18 bottles of soda to exceed $60.

Q: How many bags of popcorn and bottles of soda must I buy to exceed $60 if I want to buy an equal number of each?

A: Let's say we want to buy x bags of popcorn and x bottles of soda. The total cost of the popcorn bags and soda bottles must be greater than $60. We can set up the inequality:

6x + 3.50x > 60

Combine like terms:

9.50x > 60

Divide both sides of the inequality by 9.50:

x > 6.32

This means that we must buy at least 7 bags of popcorn and 7 bottles of soda to exceed $60 if we want to buy an equal number of each.

Q: Can I buy a combination of bags of popcorn and bottles of soda that will exactly equal $60?

A: Let's say we want to buy x bags of popcorn and y bottles of soda. The total cost of the popcorn bags and soda bottles must be exactly $60. We can set up the equation:

6x + 3.50y = 60

This is a linear equation, and we can solve for x and y using algebra. However, we will not get a whole number solution for x and y. This means that we cannot buy a combination of bags of popcorn and bottles of soda that will exactly equal $60.

Q: What if I want to buy a combination of bags of popcorn and bottles of soda that will exceed $60 by a certain amount?

A: Let's say we want to buy x bags of popcorn and y bottles of soda, and we want the total cost to exceed $60 by a certain amount, say $10. We can set up the inequality:

6x + 3.50y > 70

We can solve this inequality using algebra, and we will get a range of possible values for x and y. This means that we can buy a combination of bags of popcorn and bottles of soda that will exceed $60 by a certain amount.

Conclusion

In conclusion, we have answered some frequently asked questions related to the problem of a movie theater's concession stand. We have found the minimum number of bags of popcorn and bottles of soda required to exceed $60, and we have explored the relationship between the number of bags of popcorn and bottles of soda. We have also discussed how to buy a combination of bags of popcorn and bottles of soda that will exceed $60 by a certain amount.