Optimizartion 5) Find the avea of the largest rectangle an be inscribed in a right triangle with lee lenytns 4cm and 5 cm' it two sides of th lie along the legs
Optimizartion 5) Find the avea of the largest rectangle an be inscribed in a right triangle with lee lenytns 4cm and 5 cm' it two sides of th lie along the legs
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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![### Optimization Problem
**Problem Statement:**
5. *Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 4 cm and 5 cm if two sides of the rectangle lie along the legs.*
**Solution Steps:**
1. **Define the Variables:**
- Let the lengths of the legs of the right triangle be \(a = 4 \text{ cm}\) and \(b = 5 \text{ cm}\).
- Let the dimensions of the rectangle be \(x \text{ cm}\) and \(y \text{ cm}\).
2. **Determine the Relationship between \(x\) and \(y\):**
- Because the right triangle has legs of lengths 4 cm and 5 cm, the relationship between \(x\) and \(y\) is given by the equation \( \frac{x}{a} + \frac{y}{b} = 1 \).
3. **Formulate the Area Function to be Maximized:**
- The area \(A\) of the rectangle is given by \(A = xy\).
4. **Substitute \(y\) in terms of \(x\) into the Area Function:**
- Using the relationship \( \frac{x}{4} + \frac{y}{5} = 1 \) to express \(y\) in terms of \(x\), we get:
\[
y = 5 \left(1 - \frac{x}{4}\right) = 5 - \frac{5x}{4}.
\]
- Substitute \(y\) into the area function:
\[
A = x \left(5 - \frac{5x}{4}\right) = 5x - \frac{5x^2}{4}.
\]
5. **Differentiate the Area Function to Find Maximum Area:**
- Differentiate \(A\) with respect to \(x\):
\[
\frac{dA}{dx} = 5 - \frac{5x}{2}.
\]
- Set the derivative equal to zero to find the critical points:
\[
5 - \frac{5x}{2} = 0 \implies x = 2.
\]
6. **Determine the Corresponding \(y\) Value:**
- Substitute \(x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38aaaeb0-3205-4fdb-bd84-df06c53da4aa%2Ffc8914cf-301e-473b-a7fa-106a30775f35%2F6q0b5xh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Optimization Problem
**Problem Statement:**
5. *Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 4 cm and 5 cm if two sides of the rectangle lie along the legs.*
**Solution Steps:**
1. **Define the Variables:**
- Let the lengths of the legs of the right triangle be \(a = 4 \text{ cm}\) and \(b = 5 \text{ cm}\).
- Let the dimensions of the rectangle be \(x \text{ cm}\) and \(y \text{ cm}\).
2. **Determine the Relationship between \(x\) and \(y\):**
- Because the right triangle has legs of lengths 4 cm and 5 cm, the relationship between \(x\) and \(y\) is given by the equation \( \frac{x}{a} + \frac{y}{b} = 1 \).
3. **Formulate the Area Function to be Maximized:**
- The area \(A\) of the rectangle is given by \(A = xy\).
4. **Substitute \(y\) in terms of \(x\) into the Area Function:**
- Using the relationship \( \frac{x}{4} + \frac{y}{5} = 1 \) to express \(y\) in terms of \(x\), we get:
\[
y = 5 \left(1 - \frac{x}{4}\right) = 5 - \frac{5x}{4}.
\]
- Substitute \(y\) into the area function:
\[
A = x \left(5 - \frac{5x}{4}\right) = 5x - \frac{5x^2}{4}.
\]
5. **Differentiate the Area Function to Find Maximum Area:**
- Differentiate \(A\) with respect to \(x\):
\[
\frac{dA}{dx} = 5 - \frac{5x}{2}.
\]
- Set the derivative equal to zero to find the critical points:
\[
5 - \frac{5x}{2} = 0 \implies x = 2.
\]
6. **Determine the Corresponding \(y\) Value:**
- Substitute \(x
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