Find the maximum and minimum volumes of a rectangular box whose surface area is 1500 cm² and whose total edge length is 200 cm.
Find the maximum and minimum volumes of a rectangular box whose surface area is 1500 cm² and whose total edge length is 200 cm.
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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![**Problem 3: Optimization of a Rectangular Box**
Find the maximum and minimum volumes of a rectangular box whose surface area is \(1500 \, \text{cm}^2\) and whose total edge length is \(200 \, \text{cm}\).
**Explanation:**
To solve this problem, you'll need to use the formulas for surface area and edge length of a rectangular box. The box has dimensions \(x\), \(y\), and \(z\).
1. **Surface Area Formula:**
\[
2(xy + yz + zx) = 1500
\]
2. **Total Edge Length Formula:**
\[
4(x + y + z) = 200
\]
From this, you can derive relationships between the dimensions and use calculus to find the maximum and minimum volumes, represented by \(V = xyz\).
**Steps:**
1. Solve the edge length equation for one variable, say \(z\), in terms of \(x\) and \(y\).
2. Substitute \(z\) in the surface area equation to express the surface area in terms of \(x\) and \(y\).
3. Find the critical points by differentiating the volume function \(V(x, y)\) with respect to \(x\) and \(y\).
4. Use the second derivative test or analyze the sign changes of the derivatives to determine which points correspond to maximum and minimum volumes.
**Note:** This type of problem involves the application of Lagrange multipliers or systems of equations to find the extrema under constraints.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7538180c-5348-4d54-a5c1-c840ed6fc5b5%2Fee2a78aa-ab21-4693-a58c-44019712d065%2F9miv4tc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3: Optimization of a Rectangular Box**
Find the maximum and minimum volumes of a rectangular box whose surface area is \(1500 \, \text{cm}^2\) and whose total edge length is \(200 \, \text{cm}\).
**Explanation:**
To solve this problem, you'll need to use the formulas for surface area and edge length of a rectangular box. The box has dimensions \(x\), \(y\), and \(z\).
1. **Surface Area Formula:**
\[
2(xy + yz + zx) = 1500
\]
2. **Total Edge Length Formula:**
\[
4(x + y + z) = 200
\]
From this, you can derive relationships between the dimensions and use calculus to find the maximum and minimum volumes, represented by \(V = xyz\).
**Steps:**
1. Solve the edge length equation for one variable, say \(z\), in terms of \(x\) and \(y\).
2. Substitute \(z\) in the surface area equation to express the surface area in terms of \(x\) and \(y\).
3. Find the critical points by differentiating the volume function \(V(x, y)\) with respect to \(x\) and \(y\).
4. Use the second derivative test or analyze the sign changes of the derivatives to determine which points correspond to maximum and minimum volumes.
**Note:** This type of problem involves the application of Lagrange multipliers or systems of equations to find the extrema under constraints.
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