(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...
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![**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.
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