Answer :
To determine the number of spas and solar heaters that should be sold to maximize the total revenue, we need to find the critical points of the given revenue function \( R(x, y) \). Additionally, the second-order partial derivatives are required to classify the nature of these critical points.
### Step-by-Step Solution:
1. Revenue Function:
[tex]\[ R(x, y) = 11 + 228x + 234y - 6x^2 - 9y^2 - 6xy \][/tex]
2. First-Order Partial Derivatives:
To find the critical points, we need to take the partial derivatives with respect to \(x\) and \(y\) and set them to zero.
[tex]\[ \frac{\partial R}{\partial x} = 228 - 12x - 6y \][/tex]
[tex]\[ \frac{\partial R}{\partial y} = 234 - 18y - 6x \][/tex]
Setting these partial derivatives to zero:
[tex]\[ 228 - 12x - 6y = 0 \quad (1) \][/tex]
[tex]\[ 234 - 18y - 6x = 0 \quad (2) \][/tex]
3. Solve the System of Equations:
From equation (1):
[tex]\[ 228 - 12x - 6y = 0 \][/tex]
[tex]\[ 2x + y = 38 \quad (3) \][/tex]
From equation (2):
[tex]\[ 234 - 18y - 6x = 0 \][/tex]
[tex]\[ x + 3y = 39 \quad (4) \][/tex]
Solving equations (3) and (4) simultaneously, we find:
[tex]\[ 2x + y = 38 \][/tex]
[tex]\[ x + 3y = 39 \][/tex]
Solving these, we get:
[tex]\[ x = 15, \quad y = 8 \][/tex]
4. Second-Order Partial Derivatives:
To classify the critical point, we compute the second-order partial derivatives:
[tex]\[ \frac{\partial^2 R}{\partial x^2} = R_{xx} = -12 \][/tex]
[tex]\[ \frac{\partial^2 R}{\partial y^2} = R_{yy} = -18 \][/tex]
[tex]\[ \frac{\partial^2 R}{\partial x \partial y} = R_{xy} = -6 \][/tex]
5. Determine the Nature of the Critical Point:
The second partial derivative test involves the following determinant \(D\):
[tex]\[ D = R_{xx} R_{yy} - (R_{xy})^2 \][/tex]
Substituting the values:
[tex]\[ D = (-12)(-18) - (-6)^2 \][/tex]
[tex]\[ D = 216 - 36 = 180 \][/tex]
Since \(D > 0\) and \(R_{xx} < 0\), the critical point \((15, 8)\) is a local maximum.
6. Maximum Revenue:
Substitute the critical point back into the revenue function to find the maximum revenue:
[tex]\[ R(15, 8) = 11 + 228(15) + 234(8) - 6(15)^2 - 9(8)^2 - 6(15)(8) \][/tex]
[tex]\[ = 11 + 3420 + 1872 - 1350 - 576 - 720 \][/tex]
[tex]\[ = 2657 \text{ (hundreds of dollars)} = 265,700 \text{ dollars} \][/tex]
### Final Answers:
- The number of spas and solar heaters that should be sold to maximize revenue are \(15\) and \(8\) respectively.
- The maximum revenue is 2657 (in hundreds of dollars), which equals 265,700 dollars.
- The second-order partial derivatives are:
[tex]\[ R_{xx} = -12, \quad R_{yy} = -18, \quad R_{xy} = -6 \][/tex]
### Step-by-Step Solution:
1. Revenue Function:
[tex]\[ R(x, y) = 11 + 228x + 234y - 6x^2 - 9y^2 - 6xy \][/tex]
2. First-Order Partial Derivatives:
To find the critical points, we need to take the partial derivatives with respect to \(x\) and \(y\) and set them to zero.
[tex]\[ \frac{\partial R}{\partial x} = 228 - 12x - 6y \][/tex]
[tex]\[ \frac{\partial R}{\partial y} = 234 - 18y - 6x \][/tex]
Setting these partial derivatives to zero:
[tex]\[ 228 - 12x - 6y = 0 \quad (1) \][/tex]
[tex]\[ 234 - 18y - 6x = 0 \quad (2) \][/tex]
3. Solve the System of Equations:
From equation (1):
[tex]\[ 228 - 12x - 6y = 0 \][/tex]
[tex]\[ 2x + y = 38 \quad (3) \][/tex]
From equation (2):
[tex]\[ 234 - 18y - 6x = 0 \][/tex]
[tex]\[ x + 3y = 39 \quad (4) \][/tex]
Solving equations (3) and (4) simultaneously, we find:
[tex]\[ 2x + y = 38 \][/tex]
[tex]\[ x + 3y = 39 \][/tex]
Solving these, we get:
[tex]\[ x = 15, \quad y = 8 \][/tex]
4. Second-Order Partial Derivatives:
To classify the critical point, we compute the second-order partial derivatives:
[tex]\[ \frac{\partial^2 R}{\partial x^2} = R_{xx} = -12 \][/tex]
[tex]\[ \frac{\partial^2 R}{\partial y^2} = R_{yy} = -18 \][/tex]
[tex]\[ \frac{\partial^2 R}{\partial x \partial y} = R_{xy} = -6 \][/tex]
5. Determine the Nature of the Critical Point:
The second partial derivative test involves the following determinant \(D\):
[tex]\[ D = R_{xx} R_{yy} - (R_{xy})^2 \][/tex]
Substituting the values:
[tex]\[ D = (-12)(-18) - (-6)^2 \][/tex]
[tex]\[ D = 216 - 36 = 180 \][/tex]
Since \(D > 0\) and \(R_{xx} < 0\), the critical point \((15, 8)\) is a local maximum.
6. Maximum Revenue:
Substitute the critical point back into the revenue function to find the maximum revenue:
[tex]\[ R(15, 8) = 11 + 228(15) + 234(8) - 6(15)^2 - 9(8)^2 - 6(15)(8) \][/tex]
[tex]\[ = 11 + 3420 + 1872 - 1350 - 576 - 720 \][/tex]
[tex]\[ = 2657 \text{ (hundreds of dollars)} = 265,700 \text{ dollars} \][/tex]
### Final Answers:
- The number of spas and solar heaters that should be sold to maximize revenue are \(15\) and \(8\) respectively.
- The maximum revenue is 2657 (in hundreds of dollars), which equals 265,700 dollars.
- The second-order partial derivatives are:
[tex]\[ R_{xx} = -12, \quad R_{yy} = -18, \quad R_{xy} = -6 \][/tex]
