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. and Rxy Find the derivatives Rxx, Ryy Rxx Ryy Rxy =
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. and Rxy Find the derivatives Rxx, Ryy Rxx Ryy Rxy =
Chapter2: Functions And Their Graphs
Section2.4: A Library Of Parent Functions
Problem 47E: During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate...
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![**Problem Statement**
The total revenue (in hundreds of dollars) from the sale of \( x \) spas and \( y \) solar heaters is approximated by the function:
\[ R(x,y) = 16 + 329x + 275y - 8x^2 - 6y^2 - 11xy. \]
**Objective**
1. Find the number of spas (\( x \)) and solar heaters (\( y \)) that should be sold to produce maximum revenue.
2. Find the maximum revenue.
---
**Step 1: Calculating Second Order Partial Derivatives**
Find the partial derivatives \( R_{xx} \), \( R_{yy} \), and \( R_{xy} \).
\[ R_{xx} = \boxed{ } \]
\[ R_{yy} = \boxed{ } \]
\[ R_{xy} = \boxed{ } \]
(Note: Students will compute the second-order partial derivatives and fill in the boxes.)
---
**Explanation of Mathematical Terms**
- **Second Order Partial Derivatives**: These are derivatives of the function with respect to each variable, taken twice. They help determine the concavity of the function and are essential in finding local maxima and minima.
- \( R_{xx} \): The second partial derivative of \( R \) with respect to \( x \).
- \( R_{yy} \): The second partial derivative of \( R \) with respect to \( y \).
- \( R_{xy} \): The mixed partial derivative of \( R \) with respect to \( x \) and \( y \).
Understanding these derivatives helps in applying the second derivative test for identifying the maximum or minimum points for the revenue function.
---
**Graph and Solution Approach**
While the exact graphical representation is not included here, students can visualize the revenue function \( R(x, y) \) as a 3D surface where the height represents revenue. The task involves finding the peak of this surface (i.e., the point corresponding to the maximum revenue).
**Graph Interpretation**
1. **3D Surface Plot**: It shows how \( R \) changes with \( x \) and \( y \).
2. **Contours**: Contour lines can assist in visualizing peaks and troughs in the revenue function.
**Solution Strategy**
1. **Find First Order Partial Derivatives**: \( R_x \) and \( R_y \).
2. **Set First](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F756689fc-aa25-4045-a342-f553545a40f2%2Fd28af854-ab3c-4f59-ad7a-69b755839583%2Fuzfu0ej_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
The total revenue (in hundreds of dollars) from the sale of \( x \) spas and \( y \) solar heaters is approximated by the function:
\[ R(x,y) = 16 + 329x + 275y - 8x^2 - 6y^2 - 11xy. \]
**Objective**
1. Find the number of spas (\( x \)) and solar heaters (\( y \)) that should be sold to produce maximum revenue.
2. Find the maximum revenue.
---
**Step 1: Calculating Second Order Partial Derivatives**
Find the partial derivatives \( R_{xx} \), \( R_{yy} \), and \( R_{xy} \).
\[ R_{xx} = \boxed{ } \]
\[ R_{yy} = \boxed{ } \]
\[ R_{xy} = \boxed{ } \]
(Note: Students will compute the second-order partial derivatives and fill in the boxes.)
---
**Explanation of Mathematical Terms**
- **Second Order Partial Derivatives**: These are derivatives of the function with respect to each variable, taken twice. They help determine the concavity of the function and are essential in finding local maxima and minima.
- \( R_{xx} \): The second partial derivative of \( R \) with respect to \( x \).
- \( R_{yy} \): The second partial derivative of \( R \) with respect to \( y \).
- \( R_{xy} \): The mixed partial derivative of \( R \) with respect to \( x \) and \( y \).
Understanding these derivatives helps in applying the second derivative test for identifying the maximum or minimum points for the revenue function.
---
**Graph and Solution Approach**
While the exact graphical representation is not included here, students can visualize the revenue function \( R(x, y) \) as a 3D surface where the height represents revenue. The task involves finding the peak of this surface (i.e., the point corresponding to the maximum revenue).
**Graph Interpretation**
1. **3D Surface Plot**: It shows how \( R \) changes with \( x \) and \( y \).
2. **Contours**: Contour lines can assist in visualizing peaks and troughs in the revenue function.
**Solution Strategy**
1. **Find First Order Partial Derivatives**: \( R_x \) and \( R_y \).
2. **Set First
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