Minimizing the Objective Function c60x 85y
What is the Objective Function?
The objective function is a mathematical expression that represents the quantity or value that we want to minimize or maximize. In this case, the objective function is c60x 85y, where c is a constant and x and y are variables. Our goal is to find the ordered pair (x, y) that minimizes the value of c.
Visualizing the Objective Function
The objective function c60x 85y can be represented graphically as a parabola. The parabola opens upwards, indicating that the value of c increases as x and y increase. The vertex of the parabola is the point where the value of c is minimized.
Equations of the Objective Function
To minimize the objective function, we need to find the point on the parabola where the slope of the tangent line is equal to zero.
slope = d/dx (c60x 85y)
= 60 85 / (x^2 85)
= 60 / x^2
Setting the slope equal to zero, we get:
60 / x^2 = 0
x^2 = 60
x = ±√60
Since x cannot be negative, we discard the negative solution and take x = √60.
Substituting x into the Objective Function
Substituting x = √60 into the objective function c60x 85y, we get:
c60(√60) 85y
Using a calculator to simplify the expression, we get:
c60(8.99) 85y
c482.7y
H2: Finding the Ordered Pair
The ordered pair (x, y) that minimizes the objective function c60x 85y is (√60, y). We can find y by plugging x = √60 into the objective function:
c60(√60) 85y
= c60(8.99) 85y
c482.7y
y = c482.7 / 85
y ≈ 5.65
Therefore, the ordered pair (x, y) that minimizes the objective function c60x 85y is (√60, 5.65).
Important Considerations
- The vertex of the parabola represents the point where the value of c is minimized.
- The slope of the tangent line to the parabola is equal to zero at the vertex.
- The point (√60, 5.65) is the only point on the parabola where the value of c is minimized.
- We cannot minimize the objective function c60x 85y outside of this point.
Real-World Applications
The objective function c60x 85y is commonly used in real-world applications, such as economics, finance, and engineering. For example, in the calculation of the cost of goods produced, the objective function c60x 85y represents the total cost of production. By minimizing the objective function, manufacturers can optimize their production process and reduce costs.
Conclusion
In conclusion, minimizing the objective function c60x 85y requires finding the point on the parabola where the slope of the tangent line is equal to zero. The ordered pair (√60, 5.65) represents the point where the value of c is minimized. We can apply this concept to real-world applications, such as economics and finance, to optimize production processes and reduce costs.