Dan Bought $x$ Pounds Of Potatoes For \[$\$0.85\$\] Per Pound And $y$ Pounds Of Grapes For \[$\$1.29\$\] Per Pound. The Total Cost Was Less Than \[$\$5\$\]. Which Inequality Represents His Purchase?A.
Introduction
In this article, we will delve into the world of mathematical modeling, where we use mathematical concepts to describe real-world scenarios. We will explore a problem involving Dan's purchase of potatoes and grapes, and create an inequality to represent his total cost. This problem requires us to apply our understanding of algebraic expressions and inequalities to model a real-world situation.
Understanding the Problem
Dan bought pounds of potatoes for {$0.85$}$ per pound and pounds of grapes for {$1.29$}$ per pound. The total cost was less than {$5$}$. We need to create an inequality to represent this situation.
Creating the Inequality
To create the inequality, we need to calculate the total cost of the potatoes and grapes. The cost of the potatoes is given by , and the cost of the grapes is given by . The total cost is the sum of these two costs, which is . Since the total cost is less than {$5$}$, we can write the inequality as:
Simplifying the Inequality
We can simplify the inequality by dividing both sides by the greatest common divisor of the coefficients, which is 0.01. This gives us:
Interpretation of the Inequality
The inequality represents the situation where Dan's total cost is less than {$5$}$. This means that the sum of the cost of the potatoes and grapes is less than {$5$}$. We can use this inequality to determine the possible values of and that satisfy the condition.
Graphical Representation
We can represent the inequality graphically by plotting the line and shading the region below it. This will give us a visual representation of the possible values of and that satisfy the inequality.
Conclusion
In this article, we created an inequality to represent Dan's purchase of potatoes and grapes. We applied our understanding of algebraic expressions and inequalities to model a real-world scenario. The inequality represents the situation where Dan's total cost is less than {$5$}$. We can use this inequality to determine the possible values of and that satisfy the condition.
Real-World Applications
This problem has real-world applications in various fields, such as economics, finance, and business. For example, a store owner may use this inequality to determine the maximum amount of money they can spend on potatoes and grapes while staying within their budget.
Future Research Directions
This problem can be extended to more complex scenarios, such as multiple items with different prices and quantities. We can also explore the use of linear programming to optimize the purchase of potatoes and grapes.
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "Linear Algebra and Its Applications" by Gilbert Strang
Appendix
Solution to the Problem
To solve the problem, we need to find the values of and that satisfy the inequality . We can use the graphical representation to determine the possible values of and .
Graphical Representation
We can represent the inequality graphically by plotting the line and shading the region below it. This will give us a visual representation of the possible values of and that satisfy the inequality.
Code
import matplotlib.pyplot as plt
import numpy as np
# Define the coefficients
a = 85
b = 129
c = 500
# Create a grid of x and y values
x = np.linspace(0, 10, 100)
y = np.linspace(0, 10, 100)
X, Y = np.meshgrid(x, y)
# Calculate the value of the inequality
Z = a * X + b * Y - c
# Plot the line
plt.plot(X[0, :], a * X[0, :] + c, 'k-')
# Shade the region below the line
plt.contourf(X, Y, Z, levels=[-1, 0], colors='b')
# Set the axis limits
plt.xlim(0, 10)
plt.ylim(0, 10)
# Show the plot
plt.show()
Introduction
In our previous article, we created an inequality to represent Dan's purchase of potatoes and grapes. We applied our understanding of algebraic expressions and inequalities to model a real-world scenario. In this article, we will answer some frequently asked questions about the inequality and provide additional insights into the problem.
Q: What is the purpose of the inequality?
A: The purpose of the inequality is to represent the situation where Dan's total cost is less than {$5$}$. This means that the sum of the cost of the potatoes and grapes is less than {$5$}$.
Q: How do I use the inequality to determine the possible values of x and y?
A: To use the inequality to determine the possible values of x and y, you can plot the line 85x + 129y = 500 and shade the region below it. This will give you a visual representation of the possible values of x and y that satisfy the inequality.
Q: What is the significance of the coefficients 85 and 129 in the inequality?
A: The coefficients 85 and 129 represent the cost of the potatoes and grapes per pound, respectively. These coefficients are used to calculate the total cost of the potatoes and grapes.
Q: Can I use the inequality to determine the maximum amount of money I can spend on potatoes and grapes?
A: Yes, you can use the inequality to determine the maximum amount of money you can spend on potatoes and grapes. Simply set the inequality to an equality (85x + 129y = 500) and solve for x and y.
Q: How do I extend this problem to more complex scenarios, such as multiple items with different prices and quantities?
A: To extend this problem to more complex scenarios, you can use linear programming to optimize the purchase of potatoes and grapes. This involves setting up a system of linear equations and inequalities to represent the constraints of the problem.
Q: What are some real-world applications of this problem?
A: Some real-world applications of this problem include:
- Economics: This problem can be used to model the behavior of consumers in a market.
- Finance: This problem can be used to determine the maximum amount of money that can be spent on a particular item.
- Business: This problem can be used to optimize the purchase of goods and services.
Q: What are some future research directions for this problem?
A: Some future research directions for this problem include:
- Extending the problem to more complex scenarios, such as multiple items with different prices and quantities.
- Using linear programming to optimize the purchase of potatoes and grapes.
- Developing new mathematical models to represent the behavior of consumers in a market.
Conclusion
In this article, we answered some frequently asked questions about the inequality and provided additional insights into the problem. We also discussed some real-world applications and future research directions for the problem. We hope that this article has been helpful in understanding the inequality and its significance.
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "Linear Algebra and Its Applications" by Gilbert Strang
Appendix
Solution to the Problem
To solve the problem, we need to find the values of x and y that satisfy the inequality 85x + 129y < 500. We can use the graphical representation to determine the possible values of x and y.
Graphical Representation
We can represent the inequality graphically by plotting the line 85x + 129y = 500 and shading the region below it. This will give us a visual representation of the possible values of x and y that satisfy the inequality.
Code
import matplotlib.pyplot as plt
import numpy as np
# Define the coefficients
a = 85
b = 129
c = 500
# Create a grid of x and y values
x = np.linspace(0, 10, 100)
y = np.linspace(0, 10, 100)
X, Y = np.meshgrid(x, y)
# Calculate the value of the inequality
Z = a * X + b * Y - c
# Plot the line
plt.plot(X[0, :], a * X[0, :] + c, 'k-')
# Shade the region below the line
plt.contourf(X, Y, Z, levels=[-1, 0], colors='b')
# Set the axis limits
plt.xlim(0, 10)
plt.ylim(0, 10)
# Show the plot
plt.show()
This code will generate a graphical representation of the inequality, which can be used to determine the possible values of x and y that satisfy the condition.