Corners of equal size are cut from a square with sides of length 8 meters (see figure). x 8 (a) Write the area A of the resulting figure as a function of x. A = 64- -2x² Determine the domain of the function. (Enter your answer using interval notation.) [0,4] (b) Use a graphing utility to graph the area function over its domain. Use the graph to find the range of the function. (Enter your answer using interval notation.) [32, 64] Xx
Corners of equal size are cut from a square with sides of length 8 meters (see figure). x 8 (a) Write the area A of the resulting figure as a function of x. A = 64- -2x² Determine the domain of the function. (Enter your answer using interval notation.) [0,4] (b) Use a graphing utility to graph the area function over its domain. Use the graph to find the range of the function. (Enter your answer using interval notation.) [32, 64] Xx
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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The figure displays a square with each side measuring 8 meters. Small squares of side length \( x \) are cut out from each corner of the larger square. The resulting shape is an octagon with sides of length \( 8 - 2x \).
**(a) Write the area \( A \) of the resulting figure as a function of \( x \).**
Equation:
\[ A = 64 - 2x^2 \]
**(b) Determine the domain of the function. (Enter your answer using interval notation.)**
Domain:
\[ [0, 4] \]
**(c) Use a graphing utility to graph the area function over its domain. Use the graph to find the range of the function. (Enter your answer using interval notation.)**
*Note: The range was incorrectly found in the given answer. Ensure to verify your graph to find the correct range.*
Incorrect Range:
\[ [32, 64] \]
**Explanation of the Figure:**
The figure illustrates a square with a side length of 8 meters. Each corner of the square is marked with an \( x \) to represent the corners being cut out. The sides of the resulting figure are labeled with \( 8 - 2x \), indicating the dimensions after the corners are removed.
**Steps for the Calculation:**
1. **Calculate the Area of the Square:**
\[ \text{Area of the Square} = 8 \times 8 = 64 \text{ square meters} \]
2. **Calculate the Area of One Removed Corner:**
\[ \text{Area of One Corner} = x \times x = x^2 \]
3. **Calculate the Total Area of Removed Corners:**
Since 4 corners are removed:
\[ \text{Total Area Removed} = 4 \times x^2 = 4x^2 \]
4. **Calculate the Area of the Resulting Figure:**
\[ A = \text{Area of the Square} - \text{Total Area Removed} \]
\[ A = 64 - 4x^2 \]
**Domain Determination:**
The side length \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F372618d5-d176-48ee-8ca9-30e1e1332449%2Fe674cc44-e22a-4112-bbe9-4ba3530ba14f%2Fx4mip7o_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Area Functions through Geometric Transformations**
**Corners of equal size are cut from a square with sides of length 8 meters (see figure).**
The figure displays a square with each side measuring 8 meters. Small squares of side length \( x \) are cut out from each corner of the larger square. The resulting shape is an octagon with sides of length \( 8 - 2x \).
**(a) Write the area \( A \) of the resulting figure as a function of \( x \).**
Equation:
\[ A = 64 - 2x^2 \]
**(b) Determine the domain of the function. (Enter your answer using interval notation.)**
Domain:
\[ [0, 4] \]
**(c) Use a graphing utility to graph the area function over its domain. Use the graph to find the range of the function. (Enter your answer using interval notation.)**
*Note: The range was incorrectly found in the given answer. Ensure to verify your graph to find the correct range.*
Incorrect Range:
\[ [32, 64] \]
**Explanation of the Figure:**
The figure illustrates a square with a side length of 8 meters. Each corner of the square is marked with an \( x \) to represent the corners being cut out. The sides of the resulting figure are labeled with \( 8 - 2x \), indicating the dimensions after the corners are removed.
**Steps for the Calculation:**
1. **Calculate the Area of the Square:**
\[ \text{Area of the Square} = 8 \times 8 = 64 \text{ square meters} \]
2. **Calculate the Area of One Removed Corner:**
\[ \text{Area of One Corner} = x \times x = x^2 \]
3. **Calculate the Total Area of Removed Corners:**
Since 4 corners are removed:
\[ \text{Total Area Removed} = 4 \times x^2 = 4x^2 \]
4. **Calculate the Area of the Resulting Figure:**
\[ A = \text{Area of the Square} - \text{Total Area Removed} \]
\[ A = 64 - 4x^2 \]
**Domain Determination:**
The side length \(
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