What Is The Minimum Value Of $C=7x+8y$, Given The Constraints On $x$ And $ Y Y Y [/tex] Listed Below?${ \begin{array}{l} 2x + Y \geq 8 \ x + Y \geq 6 \ x \geq 0 \ y \geq 0 \ \end{array} }$A. 32 B. 42 C.
Introduction
In linear programming, we often encounter problems that involve finding the minimum or maximum value of a function, subject to certain constraints. In this article, we will explore the problem of finding the minimum value of the function C = 7x + 8y, given the constraints on x and y listed below.
The Constraints
The constraints on x and y are as follows:
- 2x + y ≥ 8
- x + y ≥ 6
- x ≥ 0
- y ≥ 0
These constraints indicate that the values of x and y must satisfy the following conditions:
- The sum of 2x and y must be greater than or equal to 8.
- The sum of x and y must be greater than or equal to 6.
- The value of x must be greater than or equal to 0.
- The value of y must be greater than or equal to 0.
The Function C = 7x + 8y
The function C = 7x + 8y represents a linear combination of x and y. The coefficients of x and y are 7 and 8, respectively. This function can be graphed as a line in the coordinate plane.
Finding the Minimum Value of C
To find the minimum value of C, we need to find the values of x and y that satisfy the constraints and minimize the function C = 7x + 8y.
Graphical Method
One way to find the minimum value of C is to graph the constraints and the function C = 7x + 8y on the same coordinate plane. The feasible region is the area where all the constraints are satisfied.
Feasible Region
The feasible region is the area bounded by the lines 2x + y = 8, x + y = 6, x = 0, and y = 0.
Corner Points
The corner points of the feasible region are the points where the lines intersect. The corner points are (0, 8), (6, 0), and (2, 4).
Minimum Value of C
To find the minimum value of C, we need to evaluate the function C = 7x + 8y at each of the corner points.
Corner Point (0, 8)
At the corner point (0, 8), the value of C is:
C = 7(0) + 8(8) = 64
Corner Point (6, 0)
At the corner point (6, 0), the value of C is:
C = 7(6) + 8(0) = 42
Corner Point (2, 4)
At the corner point (2, 4), the value of C is:
C = 7(2) + 8(4) = 34
Conclusion
Based on the values of C at each of the corner points, we can conclude that the minimum value of C is 42, which occurs at the corner point (6, 0).
Final Answer
The final answer is 42.
Introduction
In our previous article, we explored the problem of finding the minimum value of the function C = 7x + 8y, given the constraints on x and y. In this article, we will answer some frequently asked questions (FAQs) about the minimum value of C.
Q: What are the constraints on x and y?
A: The constraints on x and y are:
- 2x + y ≥ 8
- x + y ≥ 6
- x ≥ 0
- y ≥ 0
Q: What is the function C = 7x + 8y?
A: The function C = 7x + 8y represents a linear combination of x and y. The coefficients of x and y are 7 and 8, respectively.
Q: How do we find the minimum value of C?
A: To find the minimum value of C, we need to find the values of x and y that satisfy the constraints and minimize the function C = 7x + 8y.
Q: What is the feasible region?
A: The feasible region is the area where all the constraints are satisfied. It is bounded by the lines 2x + y = 8, x + y = 6, x = 0, and y = 0.
Q: What are the corner points of the feasible region?
A: The corner points of the feasible region are the points where the lines intersect. The corner points are (0, 8), (6, 0), and (2, 4).
Q: How do we find the minimum value of C at each corner point?
A: To find the minimum value of C at each corner point, we need to evaluate the function C = 7x + 8y at each point.
Q: What is the minimum value of C?
A: Based on the values of C at each of the corner points, we can conclude that the minimum value of C is 42, which occurs at the corner point (6, 0).
Q: Why is the minimum value of C important?
A: The minimum value of C is important because it represents the smallest possible value of the function C = 7x + 8y, subject to the given constraints.
Q: How can we use the minimum value of C in real-world applications?
A: The minimum value of C can be used in real-world applications such as optimization problems, where we need to minimize a function subject to certain constraints.
Q: Can we use other methods to find the minimum value of C?
A: Yes, we can use other methods such as the graphical method, the simplex method, or the gradient descent method to find the minimum value of C.
Conclusion
In this article, we have answered some frequently asked questions (FAQs) about the minimum value of C = 7x + 8y. We have discussed the constraints on x and y, the function C = 7x + 8y, and the feasible region. We have also found the minimum value of C and discussed its importance and applications.
Final Answer
The final answer is 42.