If x thousand dollars is spent on labor and y thousand dollars is spent on euqpment,The output of a certain factory will be Q(x,y)=2x^2+y^2+6y units.If 120,000 dollars is available,how should this be allocated between labor and equipment
to generate the largest possible output
for max Q, we need
∂Q/∂x=0 and ∂Q∂y=0
∂Q/∂x = 4x
∂Q/∂y = 2y+6
so, there's a max/min at (0,-3).
Since the 2nd partials are both positive, it's a minimum.
So, we need to maximize
2x^2+y^2+6y subject to
x+y = 120
Q(0,120) = 15120
Q(120,0) = 28800
Q(119,1)
Clearly, since 2x^2 decreases faster than y^2+6y increases for each dollar moved from x to y, the max is Q(120,0).
Doesn't seem right, since some equipment must be used.
To determine how to allocate the $120,000 between labor and equipment to generate the largest possible output, we need to find the values of x and y that maximize the function Q(x, y) = 2x^2 + y^2 + 6y.
One approach to finding the maximum value is to use calculus and find the critical points of the function by taking partial derivatives with respect to x and y. However, this can be complex and time-consuming.
An alternative approach is to analyze the function graphically. We can visualize the function Q(x, y) as a surface plot in a three-dimensional coordinate system. The peaks on this surface represent the maximum points of the function.
We can use software or graphing tools to plot the function Q(x, y) = 2x^2 + y^2 + 6y and visually identify the maximum point. Here's a step-by-step guide on how to do it:
1. Plot the function: Set up a graphing tool or software that allows you to plot three-dimensional functions. Enter the function Q(x, y) = 2x^2 + y^2 + 6y.
2. Define the range: Set reasonable ranges for the values of x and y that make sense in the context of the problem. For instance, if the problem assumes thousands of dollars, you can set ranges such as x ∈ [0, 120] (in thousands) and y ∈ [0, 120] (in thousands).
3. Generate the surface plot: Generate a three-dimensional surface plot of the function Q(x, y) using the defined range of x and y.
4. Identify the maximum point: Examine the graph and locate the highest point on the surface. This point represents the maximum output for the given budget.
5. Determine the corresponding x and y values: Read off the x and y values at the maximum point on the graph.
6. Calculate budget allocation: Given that the total budget is $120,000, allocate the amount between labor (x) and equipment (y) based on the identified x and y values.
Remember that this approach provides an approximate solution and the graph should be labeled with the correct units to reflect the budget values.