The monthly profit (in dollars) of Bond and Barker Department Store depends on the level of inventory x (in thousands of dollars) and the floor space y (in thousands of square feet) available for display of the merchandise, as given by the equation below.
P(x, y) = -0.016x^2 - 17y^2 + xy + 28x + 22y - 20000
(a) Compute Px and Py when x = 4000 and y = 150.
Px = ?
Py = ?
(b) Repeat with x = 5000 and y = 150.
Px = ?
Py = ?
NOTE: I do not know how to do it. It will help if you show step by step. Thanks
a)
dp/dx = -.032x + y + 28
= -.032(4000) + 150(1) + 28
= -128 + 150 + 28
= 50
dp/dy = -34y + x + 22
= -34(150) + 4000(1) + 22
= -5100 + 4000+ 22
= -1078
b)
dp/dx = -.032x + y + 28
= -.032(5000) + 150(1) + 28
= -160 + 150 + 28
= 18
dp/dy = -34y + x + 22
= -34(150) + 5000(1) + 22
= -5100 + 5000 + 22
= -78
To compute Px and Py, we need to find the partial derivatives of the profit function P(x, y) with respect to x and y, respectively.
(a) Computing Px and Py when x = 4000 and y = 150:
Step 1: Start with the given profit function:
P(x, y) = -0.016x^2 - 17y^2 + xy + 28x + 22y - 20000
Step 2: Find Px by differentiating P(x, y) with respect to x while treating y as a constant:
Px = dP/dx = d/dx (-0.016x^2 - 17y^2 + xy + 28x + 22y - 20000)
To differentiate each term:
d/dx (-0.016x^2) = -0.016 * 2x = -0.032x
d/dx (17y^2) = 0 (since y is a constant)
d/dx (xy) = y
d/dx (28x) = 28
d/dx (22y) = 0 (since y is a constant)
d/dx (-20000) = 0 (since it is a constant)
Now, combine all the derivatives:
Px = -0.032x + y + 28
Substituting x = 4000 and y = 150:
Px = -0.032(4000) + 150 + 28
Px = -128 + 150 + 28
Px = 50
Therefore, Px = 50 when x = 4000 and y = 150.
Step 3: Find Py by differentiating P(x, y) with respect to y while treating x as a constant:
Py = dP/dy = d/dy (-0.016x^2 - 17y^2 + xy + 28x + 22y - 20000)
To differentiate each term:
d/dy (-0.016x^2) = 0 (since x is a constant)
d/dy (17y^2) = 17 * 2y = 34y
d/dy (xy) = x
d/dy (28x) = 0 (since x is a constant)
d/dy (22y) = 22
d/dy (-20000) = 0 (since it is a constant)
Now, combine all the derivatives:
Py = 34y + x + 22
Substituting x = 4000 and y = 150:
Py = 34(150) + 4000 + 22
Py = 5100 + 4000 + 22
Py = 9122
Therefore, Py = 9122 when x = 4000 and y = 150.
(b) Repeat with x = 5000 and y = 150:
Using the same profit function P(x, y) = -0.016x^2 - 17y^2 + xy + 28x + 22y - 20000, you can go through Steps 2 and 3 using the provided values x = 5000 and y = 150 to find the values of Px and Py.
Substituting x = 5000 and y = 150 in the expressions we derived in Steps 2 and 3:
For Px:
Px = -0.032(5000) + 150 + 28
Px = -160 + 150 + 28
Px = 18
Therefore, Px = 18 when x = 5000 and y = 150.
For Py:
Py = 34(150) + 5000 + 22
Py = 5100 + 5000 + 22
Py = 10122
Therefore, Py = 10122 when x = 5000 and y = 150.
I hope this step-by-step explanation helps you understand how to compute Px and Py in this particular scenario. Let me know if you need any further clarification.