(5). Find the area of the largest rectangle that can be inscribed by the region bound by the 4-x graph of f(x)= and the coordinate axes in the first quadrant. What is the maximum 2+x area? What are the dimensions of the rectangle? 0
(5). Find the area of the largest rectangle that can be inscribed by the region bound by the 4-x graph of f(x)= and the coordinate axes in the first quadrant. What is the maximum 2+x area? What are the dimensions of the rectangle? 0
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
![**Maximizing the Area of an Inscribed Rectangle**
**Problem Statement:**
Find the area of the largest rectangle that can be inscribed by the region bound by the graph of \( f(x) = \frac{4 - x}{2 + x} \) and the coordinate axes in the first quadrant. What is the maximum area? What are the dimensions of the rectangle?
**Graph Explanation:**
The graph given shows the function \( f(x) = \frac{4 - x}{2 + x} \) plotted in the first quadrant. The function is a hyperbola that starts high on the y-axis and curves downward as x increases. There is a shaded purple rectangle inscribed under this curve, touching both the y-axis and the x-axis, and extending upward to meet the curve at a certain point.
**Steps to Solve:**
1. Determine the dimensions of the rectangle in terms of \( x \).
2. Use the function to express the height of the rectangle.
3. Set up the area function in terms of \( x \).
4. Use calculus (take the derivative and set it to zero) to find the point at which the area is maximized.
5. Verify the dimensions and compute the maximum area.
**Mathematical Formulation:**
1. The width of the rectangle is \( x \).
2. The height of the rectangle is \( f(x) = \frac{4 - x}{2 + x} \).
3. The area \( A(x) \) of the rectangle can be expressed as:
\[ A(x) = x \cdot f(x) = x \cdot \frac{4 - x}{2 + x} \]
4. To maximize \( A(x) \), take the derivative \( A'(x) \) and set it to zero:
\[ A'(x) = \text{0} \]
5. Solve for \( x \) to find the dimensions giving the maximum area.
**Conclusion:**
By finding the appropriate value of \( x \), one can determine the maximum inscribed rectangle's area and dimensions. This problem involves optimization techniques commonly covered in higher-level calculus courses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99284a24-5b6d-4c49-8bfd-674d8761bb54%2F41837186-1476-4101-ac50-1a8d60bcb4ce%2F1kuqv1h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Maximizing the Area of an Inscribed Rectangle**
**Problem Statement:**
Find the area of the largest rectangle that can be inscribed by the region bound by the graph of \( f(x) = \frac{4 - x}{2 + x} \) and the coordinate axes in the first quadrant. What is the maximum area? What are the dimensions of the rectangle?
**Graph Explanation:**
The graph given shows the function \( f(x) = \frac{4 - x}{2 + x} \) plotted in the first quadrant. The function is a hyperbola that starts high on the y-axis and curves downward as x increases. There is a shaded purple rectangle inscribed under this curve, touching both the y-axis and the x-axis, and extending upward to meet the curve at a certain point.
**Steps to Solve:**
1. Determine the dimensions of the rectangle in terms of \( x \).
2. Use the function to express the height of the rectangle.
3. Set up the area function in terms of \( x \).
4. Use calculus (take the derivative and set it to zero) to find the point at which the area is maximized.
5. Verify the dimensions and compute the maximum area.
**Mathematical Formulation:**
1. The width of the rectangle is \( x \).
2. The height of the rectangle is \( f(x) = \frac{4 - x}{2 + x} \).
3. The area \( A(x) \) of the rectangle can be expressed as:
\[ A(x) = x \cdot f(x) = x \cdot \frac{4 - x}{2 + x} \]
4. To maximize \( A(x) \), take the derivative \( A'(x) \) and set it to zero:
\[ A'(x) = \text{0} \]
5. Solve for \( x \) to find the dimensions giving the maximum area.
**Conclusion:**
By finding the appropriate value of \( x \), one can determine the maximum inscribed rectangle's area and dimensions. This problem involves optimization techniques commonly covered in higher-level calculus courses.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
Expert Answers to Latest Homework Questions
Q: what is the price-to-earnings (p/E) ratio? answer plz
Q: How much good will should altitude report in its post combination consolidated balance sheet?
Q: Need help this issue
Q: In the alchemist book what are ways Santiago evolved over time
Q: At what amount should brandon ltd.record patent Y?
Q: Need answer
Q: manufacturing cost.
Q: A company acquired a trademark on January 1, 2015, for $5 million. The trademark will be used for 10…
Q: Sol This question answer
Q: choose best answer
Q: 2 POINT
Q: Zeb Enterprises' corporate charter allows it to issue 3,000,000 shares of common stock. In its first…
Q: What is the weighted average cost of capital on these financial accounting question?
Q: The Carter Corporation's next expected dividend, D1, is $2.75; its growth rate is 5.5 percent, and…
Q: correct answer please help me general accounting
Q: On January 1, 2020, Franklin Ltd. acquired a delivery truck at a cost of $750,000. It is to be…
Q: No chatgpt pls will upvote
Q: What is the amount of joint costs assigned?
Q: Can you please give me correct answer this financial accounting question?
Q: hello teacher please solve question
Q: Help this question