The total revenue (in hundreds of dollars) from the sale of x spas and y solar heaters is approximated by R(x,y) = 16 + 329x + 275y - 8x² - 6y² - 11xy. Find the number of each that should be sold to produce maximum revenue. Find the maximum revenue. Find the derivatives Rxx, Ryy, and Rxy- Rxx= -16, Ryy = -12, Rxy = -11 Selling spas and solar heaters gives the maximum revenue of $ (Simplify your answers.)
The total revenue (in hundreds of dollars) from the sale of x spas and y solar heaters is approximated by R(x,y) = 16 + 329x + 275y - 8x² - 6y² - 11xy. Find the number of each that should be sold to produce maximum revenue. Find the maximum revenue. Find the derivatives Rxx, Ryy, and Rxy- Rxx= -16, Ryy = -12, Rxy = -11 Selling spas and solar heaters gives the maximum revenue of $ (Simplify your answers.)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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![### Maximizing Revenue for Spa and Solar Heater Sales
The total revenue (in hundreds of dollars) from the sale of \(x\) spas and \(y\) solar heaters is approximated by the function \(R(x,y)\):
\[ R(x,y) = 16 + 329x + 275y - 8x^2 - 6y^2 - 11xy \]
To determine the number of spas and solar heaters that should be sold to achieve maximum revenue, and to find the maximum revenue, we follow these steps:
1. **Finding Partial Derivatives:**
Calculate the second-order partial derivatives of \(R(x,y)\):
\[
R_{xx} = -16, \quad R_{yy} = -12, \quad R_{xy} = -11
\]
2. **Interpreting the Second-Order Conditions:**
- For a function \(f(x,y)\), if the second-order partial derivatives at a critical point \((x_0, y_0)\) satisfy the conditions of a local maximum, the Hessian matrix must be negative definite.
The given values for the second-order derivatives suggest that we proceed by checking for revenue maximization.
3. **Solving for Critical Points:**
By solving the first-order partial derivatives equations of \(R(x,y)\):
\[
\frac{\partial R}{\partial x} = 329 - 16x - 11y = 0
\]
\[
\frac{\partial R}{\partial y} = 275 - 12y - 11x = 0
\]
We find the number of spas and solar heaters to sell.
4. **Maximum Revenue Calculation:**
Substitute the critical values of \(x\) and \(y\) back into the revenue function \(R(x, y)\) to find the maximum revenue.
Let us fill in the solution details:
- Selling \(\boxed{\phantom{00}}\) spas and \(\boxed{\phantom{00}}\) solar heaters gives the maximum revenue of \(\boxed{\phantom{0000}}\).
By inputting specific values for \(x\) and \(y\) into \(R(x,y)\), we complete the process and find the concrete numbers that result in maximum revenue.
(Simplify your answers in the provided boxes to achieve the precise solutions](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd4b1be92-ebc2-4282-83f4-8930c3f3e3ba%2F573ca732-ca10-4a29-b53b-c55753d5d131%2Fcd26mck_processed.png&w=3840&q=75)
Transcribed Image Text:### Maximizing Revenue for Spa and Solar Heater Sales
The total revenue (in hundreds of dollars) from the sale of \(x\) spas and \(y\) solar heaters is approximated by the function \(R(x,y)\):
\[ R(x,y) = 16 + 329x + 275y - 8x^2 - 6y^2 - 11xy \]
To determine the number of spas and solar heaters that should be sold to achieve maximum revenue, and to find the maximum revenue, we follow these steps:
1. **Finding Partial Derivatives:**
Calculate the second-order partial derivatives of \(R(x,y)\):
\[
R_{xx} = -16, \quad R_{yy} = -12, \quad R_{xy} = -11
\]
2. **Interpreting the Second-Order Conditions:**
- For a function \(f(x,y)\), if the second-order partial derivatives at a critical point \((x_0, y_0)\) satisfy the conditions of a local maximum, the Hessian matrix must be negative definite.
The given values for the second-order derivatives suggest that we proceed by checking for revenue maximization.
3. **Solving for Critical Points:**
By solving the first-order partial derivatives equations of \(R(x,y)\):
\[
\frac{\partial R}{\partial x} = 329 - 16x - 11y = 0
\]
\[
\frac{\partial R}{\partial y} = 275 - 12y - 11x = 0
\]
We find the number of spas and solar heaters to sell.
4. **Maximum Revenue Calculation:**
Substitute the critical values of \(x\) and \(y\) back into the revenue function \(R(x, y)\) to find the maximum revenue.
Let us fill in the solution details:
- Selling \(\boxed{\phantom{00}}\) spas and \(\boxed{\phantom{00}}\) solar heaters gives the maximum revenue of \(\boxed{\phantom{0000}}\).
By inputting specific values for \(x\) and \(y\) into \(R(x,y)\), we complete the process and find the concrete numbers that result in maximum revenue.
(Simplify your answers in the provided boxes to achieve the precise solutions
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