### Area***Twelve*** feet of wire is to be used to form a square and a circle.#### (a) Express the sum of the areas of the square and the circle as a function $ A $ of a side of the square $ x $.\[ A = x^2 + \frac{(6 - 2x)^2}{\pi} \]#### (b) What is the domain of $ A $? (Enter your answer using interval notation.)\[ [0, 3] \]#### (c) Use a graphing utility to graph $ A $ on its domain.**Graphs:**1. **First Graph**: - **Axes**: The x-axis ranges from 0 to 3, and the y-axis ranges from 0 to 20. - **Curve**: The graph shows a decreasing then increasing function, representing the total area as a function of $ x $. The minimum point of this curve is near $ x = 1.5 $.2. **Second Graph**: - **Axes**: The x-axis ranges from 0 to 6, and the y-axis ranges from 0 to 18. - **Curve**: This curve is similar to the first but extends further on the x-axis, showing another decreasing then increasing pattern. The minimum point is the same, near $ x = 1.5 $, but the function increases more as $ x $ approaches 6.3. **Third Graph**: - **Axes**: The x-axis ranges from 0 to 6, and the y-axis ranges from 0 to 17. - **Curve**: This graph also shows a similar behavior with a decreasing and increasing pattern, and the minimum point is at $ x = 1.5 $ before the function increases again as $ x $ approaches 6. ### Optimizing Wire Usage for Enclosing Areas#### Graphical AnalysisIn the provided images, there are four graphs displaying different functions. These graphs are essential for interpreting and solving optimization problems involving wire to enclose areas. Let's briefly explain each graph:1. **Graph 1 (Top left)**: - **Axes**: The x-axis ranges from 0 to 3, and the y-axis is also scaled accordingly. - **Plot**: This is a curve starting at near (0,0), rising to a max value around (1.5,1.5), and then dropping again to (3,0).2. **Graph 2 (Top right)**: - **Axes**: The x-axis ranges from 0 to 6 while the y-axis is scaled to show the curve's range. - **Plot**: This is another curve starting at about (0,0), reaching a peak roughly at (3,3), and descending to (6,0).3. **Graph 3 (Bottom left)**: - **Axes**: The x-axis ranges from 0.2 to 1.4, and the y-axis from 5 to 15. - **Plot**: This parabola-like curve starts near (0.2,10), dips down to about (0.8,5), and then rises again towards (1.4,15).4. **Graph 4 (Bottom right)**: - **Axes**: The x-axis ranges from 0 to 1, and the y-axis from 5 to 15. - **Plot**: This is another curve beginning around (0,5), peaking approximately at (0.5,15), and returning to (1,5).These graphical variations must be understood to solve the optimization problems accurately.#### Optimization QuestionsNext, consider the provided problem statement regarding wire usage. It involves determining the amount of wire for enclosing the least and greatest total areas between a square and a circle.1. **Minimizing Total Area**: - The fields for wire length used for the square and the circle are meant to be filled. The example incorrectly estimates that **0.84 ft** of wire for the square doesn't yield the least area. The optimal lengths need to be recalculated.2. **Maximizing Total Area**: - There are unfilled fields for determining the wire

Side of the square=x Length of wire used for the making of square= perimeter of square=4x Length of…

Area Twelve feet of wire is to be used to form a square and a circle. (a) Express the sum of the areas of the square and the circle as a function A of a side of the square x. x² (6 – 2x)² A = (b) What is the domain of A? (Enter your answer using interval notation.) [0,3] (c) Use a graphing utility to graph A on its domain.

Area Twelve feet of wire is to be used to form a square and a circle. (a) Express the sum of the areas of the square and the circle as a function A of a side of the square x. x² (6 – 2x)² A = (b) What is the domain of A? (Enter your answer using interval notation.) [0,3] (c) Use a graphing utility to graph A on its domain.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.

Question

### Area

***Twelve*** feet of wire is to be used to form a square and a circle.

#### (a) Express the sum of the areas of the square and the circle as a function $ A $ of a side of the square $ x $.
\[ A = x^2 + \frac{(6 - 2x)^2}{\pi} \]

#### (b) What is the domain of $ A $? (Enter your answer using interval notation.)
\[ [0, 3] \]

#### (c) Use a graphing utility to graph $ A $ on its domain.

**Graphs:**

1. **First Graph**:
- **Axes**: The x-axis ranges from 0 to 3, and the y-axis ranges from 0 to 20.
- **Curve**: The graph shows a decreasing then increasing function, representing the total area as a function of $ x $. The minimum point of this curve is near $ x = 1.5 $.

2. **Second Graph**:
- **Axes**: The x-axis ranges from 0 to 6, and the y-axis ranges from 0 to 18.
- **Curve**: This curve is similar to the first but extends further on the x-axis, showing another decreasing then increasing pattern. The minimum point is the same, near $ x = 1.5 $, but the function increases more as $ x $ approaches 6.

3. **Third Graph**:
- **Axes**: The x-axis ranges from 0 to 6, and the y-axis ranges from 0 to 17.
- **Curve**: This graph also shows a similar behavior with a decreasing and increasing pattern, and the minimum point is at $ x = 1.5 $ before the function increases again as $ x $ approaches 6.

### Optimizing Wire Usage for Enclosing Areas

#### Graphical Analysis
In the provided images, there are four graphs displaying different functions. These graphs are essential for interpreting and solving optimization problems involving wire to enclose areas. Let's briefly explain each graph:

1. **Graph 1 (Top left)**:
- **Axes**: The x-axis ranges from 0 to 3, and the y-axis is also scaled accordingly.
- **Plot**: This is a curve starting at near (0,0), rising to a max value around (1.5,1.5), and then dropping again to (3,0).

2. **Graph 2 (Top right)**:
- **Axes**: The x-axis ranges from 0 to 6 while the y-axis is scaled to show the curve's range.
- **Plot**: This is another curve starting at about (0,0), reaching a peak roughly at (3,3), and descending to (6,0).

3. **Graph 3 (Bottom left)**:
- **Axes**: The x-axis ranges from 0.2 to 1.4, and the y-axis from 5 to 15.
- **Plot**: This parabola-like curve starts near (0.2,10), dips down to about (0.8,5), and then rises again towards (1.4,15).

4. **Graph 4 (Bottom right)**:
- **Axes**: The x-axis ranges from 0 to 1, and the y-axis from 5 to 15.
- **Plot**: This is another curve beginning around (0,5), peaking approximately at (0.5,15), and returning to (1,5).

These graphical variations must be understood to solve the optimization problems accurately.

#### Optimization Questions
Next, consider the provided problem statement regarding wire usage. It involves determining the amount of wire for enclosing the least and greatest total areas between a square and a circle.

1. **Minimizing Total Area**:
- The fields for wire length used for the square and the circle are meant to be filled. The example incorrectly estimates that **0.84 ft** of wire for the square doesn't yield the least area. The optimal lengths need to be recalculated.

2. **Maximizing Total Area**:
- There are unfilled fields for determining the wire

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