Carly Is Selling Cups Of Lemonade At A Stand Outside Her House. She Has Enough Supplies To Make A Maximum Of 258 Ounces Of Lemonade. A Regular Cup Holds 13 Ounces Of Lemonade, And A Small Cup Holds 6 Ounces Of Lemonade.Select The Inequality In Standard

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Carly's Lemonade Stand: A Math Problem

Carly is selling cups of lemonade at a stand outside her house. She has enough supplies to make a maximum of 258 ounces of lemonade. A regular cup holds 13 ounces of lemonade, and a small cup holds 6 ounces of lemonade. In this problem, we will explore the inequalities that represent the number of cups Carly can sell.

To solve this problem, we need to understand the constraints and the variables involved. Let's denote the number of regular cups as R and the number of small cups as S. We know that the total number of cups Carly can sell is limited by the amount of lemonade she has, which is 258 ounces.

We can represent the total amount of lemonade as an inequality. Since each regular cup holds 13 ounces and each small cup holds 6 ounces, we can write the inequality as:

13R + 6S ≤ 258

This inequality represents the constraint that the total amount of lemonade used cannot exceed 258 ounces.

To convert the inequality to standard form, we need to isolate the variable on one side of the inequality. In this case, we can subtract 6S from both sides of the inequality to get:

13R ≤ 258 - 6S

However, we want to isolate R, so we can divide both sides of the inequality by 13 to get:

R ≤ (258 - 6S) / 13

This is the standard form of the inequality.

We can simplify the inequality by multiplying both sides by 13 to get rid of the fraction:

13R ≤ 258 - 6S

This inequality is now in a simpler form, but it still represents the same constraint.

We can represent the inequality graphically by plotting the line 13R = 258 - 6S. The region below the line represents the values of R and S that satisfy the inequality.

In this problem, we explored the inequalities that represent the number of cups Carly can sell. We represented the total amount of lemonade as an inequality and converted it to standard form. We also simplified the inequality and represented it graphically. This problem demonstrates the importance of understanding inequalities and how they can be used to represent real-world constraints.

  • Inequalities can be used to represent real-world constraints.
  • The standard form of an inequality is obtained by isolating the variable on one side of the inequality.
  • Graphical representation can be used to visualize the inequality and the region that satisfies it.
  • Explore other inequalities that represent real-world constraints.
  • Practice converting inequalities to standard form and simplifying them.
  • Use graphical representation to visualize inequalities and the regions that satisfy them.

13R + 6S ≤ 258

13R ≤ 258 - 6S

The region below the line 13R = 258 - 6S represents the values of R and S that satisfy the inequality.

import numpy as np

a = 13
b = 6
c = 258



R = np.linspace(0, 20, 100)
S = np.linspace(0, 40, 100)



R, S = np.meshgrid(R, S)



inequality = a * R + b * S - c



import matplotlib.pyplot as plt


plt.contourf(R, S, inequality, levels=[0])
plt.colorbar(label='Inequality Value')
plt.xlabel('R')
plt.ylabel('S')
plt.title('Region that Satisfies the Inequality')
plt.show()


**Carly's Lemonade Stand: A Math Problem Q&A**

**Introduction**
===============

In our previous article, we explored the inequalities that represent the number of cups Carly can sell at her lemonade stand. We represented the total amount of lemonade as an inequality and converted it to standard form. We also simplified the inequality and represented it graphically. In this article, we will answer some frequently asked questions about the problem.

**Q&A**
=====

**Q: What is the maximum number of regular cups Carly can sell?**
A: To find the maximum number of regular cups Carly can sell, we need to set S = 0 in the inequality 13R + 6S ≤ 258. This gives us 13R ≤ 258, which simplifies to R ≤ 19.846. Since R must be an integer, the maximum number of regular cups Carly can sell is 19.

**Q: What is the maximum number of small cups Carly can sell?**
A: To find the maximum number of small cups Carly can sell, we need to set R = 0 in the inequality 13R + 6S ≤ 258. This gives us 6S ≤ 258, which simplifies to S ≤ 43. This means that Carly can sell a maximum of 43 small cups.

**Q: How many cups can Carly sell if she sells 10 regular cups?**
A: If Carly sells 10 regular cups, she will use 10 x 13 = 130 ounces of lemonade. This leaves her with 258 - 130 = 128 ounces of lemonade. Since each small cup holds 6 ounces of lemonade, Carly can sell 128 / 6 = 21.33 small cups. Since Carly cannot sell a fraction of a cup, she can sell 21 small cups.

**Q: How many cups can Carly sell if she sells 20 small cups?**
A: If Carly sells 20 small cups, she will use 20 x 6 = 120 ounces of lemonade. This leaves her with 258 - 120 = 138 ounces of lemonade. Since each regular cup holds 13 ounces of lemonade, Carly can sell 138 / 13 = 10.62 regular cups. Since Carly cannot sell a fraction of a cup, she can sell 10 regular cups.

**Q: What is the relationship between the number of regular cups and the number of small cups?**
A: The number of regular cups and the number of small cups are related by the inequality 13R + 6S ≤ 258. This means that as the number of regular cups increases, the number of small cups must decrease, and vice versa.

**Q: Can Carly sell any number of cups she wants?**
A: No, Carly is limited by the amount of lemonade she has, which is 258 ounces. She cannot sell more cups than she has lemonade for.

**Conclusion**
==========

In this article, we answered some frequently asked questions about Carly's lemonade stand problem. We explored the inequalities that represent the number of cups Carly can sell and converted them to standard form. We also simplified the inequalities and represented them graphically. This problem demonstrates the importance of understanding inequalities and how they can be used to represent real-world constraints.

**Key Takeaways**
================

* Inequalities can be used to represent real-world constraints.
* The standard form of an inequality is obtained by isolating the variable on one side of the inequality.
* Graphical representation can be used to visualize the inequality and the region that satisfies it.
* The number of regular cups and the number of small cups are related by the inequality 13R + 6S ≤ 258.

**Further Exploration**
=====================

* Explore other inequalities that represent real-world constraints.
* Practice converting inequalities to standard form and simplifying them.
* Use graphical representation to visualize inequalities and the regions that satisfy them.

**Inequality in Standard Form**
==========================

13R + 6S ≤ 258

**Simplified Inequality**
==========================

13R ≤ 258 - 6S

**Graphical Representation**
==========================

The region below the line 13R = 258 - 6S represents the values of R and S that satisfy the inequality.

**Code**
=====

```python
import numpy as np

# Define the coefficients of the inequality
a = 13
b = 6
c = 258

# Define the range of values for R and S
R = np.linspace(0, 20, 100)
S = np.linspace(0, 40, 100)

# Create a meshgrid of R and S values
R, S = np.meshgrid(R, S)

# Calculate the value of the inequality
inequality = a * R + b * S - c

# Plot the region that satisfies the inequality
import matplotlib.pyplot as plt

plt.contourf(R, S, inequality, levels=[0])
plt.colorbar(label='Inequality Value')
plt.xlabel('R')
plt.ylabel('S')
plt.title('Region that Satisfies the Inequality')
plt.show()
</code></pre>