Question

the total revenue (in hundreds of dollars) from the sale of x spas and y solar heaters is approximated by r(x,y)=12 + 208x + 308y - 4x² - 9y² - 8xy. find the number of each that should be sold to produce maximum revenue. find the maximum revenue.
find the derivatives r_xx, r_yy, and r_xy.
r_xx = □, r_yy = □, r_xy = □

Explanation:

Step1: Find first - order partial derivatives

The revenue function is \(R(x,y)=12 + 208x+308y - 4x^{2}-9y^{2}-8xy\).
The first - order partial derivative with respect to \(x\) is \(R_{x}=\frac{\partial R}{\partial x}=208 - 8x-8y\).
The first - order partial derivative with respect to \(y\) is \(R_{y}=\frac{\partial R}{\partial y}=308 - 18y-8x\).

Step2: Find second - order partial derivatives

The second - order partial derivative \(R_{xx}=\frac{\partial^{2}R}{\partial x^{2}}=-8\).
The second - order partial derivative \(R_{yy}=\frac{\partial^{2}R}{\partial y^{2}}=-18\).
The mixed - second - order partial derivative \(R_{xy}=\frac{\partial^{2}R}{\partial x\partial y}=-8\).

Step3: Find critical points

Set \(R_{x}=0\) and \(R_{y}=0\).
From \(R_{x}=208 - 8x - 8y = 0\), we get \(x + y=26\), so \(x = 26 - y\).
Substitute \(x = 26 - y\) into \(R_{y}=308-18y - 8x=0\).
\(308-18y-8(26 - y)=0\).
\(308-18y-208 + 8y=0\).
\(100-10y = 0\), so \(y = 10\).
Substitute \(y = 10\) into \(x = 26 - y\), we get \(x=16\).

Step4: Use the second - derivative test

The discriminant \(D=R_{xx}R_{yy}-(R_{xy})^{2}=(-8)\times(-18)-(-8)^{2}=144 - 64 = 80\).
Since \(R_{xx}=-8\lt0\) and \(D = 80\gt0\), the function \(R(x,y)\) has a maximum at the point \((x,y)=(16,10)\).

Step5: Calculate the maximum revenue

Substitute \(x = 16\) and \(y = 10\) into \(R(x,y)\):
\(R(16,10)=12+208\times16 + 308\times10-4\times16^{2}-9\times10^{2}-8\times16\times10\).
\(=12+3328+3080-4\times256-900 - 1280\).
\(=12+3328+3080 - 1024-900 - 1280\).
\(=3216\).

Answer:

\(R_{xx}=-8\), \(R_{yy}=-18\), \(R_{xy}=-8\). The number of spas \(x = 16\) (in hundreds) and solar heaters \(y = 10\) (in hundreds) should be sold. The maximum revenue is \(3216\) (in hundreds of dollars).