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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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Answer B and C
![**Problem Statement:**
Find the volume and the dimensions of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex on the surface \( z = 10 - 2x - y^2 \).
**Figures Description:**
1. The first figure is a 3D graph showing a surface in the first octant formed by the equation \( z = 10 - 2x - y^2 \). The surface appears as a curved shape coming downwards, where the height at each point is determined by the function of \( x \) and \( y \).
2. The second figure is a simple 3D coordinate system for sketching. It has labeled axes: \( x \), \( y \), and \( z \).
**Tasks:**
a) Find the volume function (in terms of \( x \) & \( y \)) for this rectangular box and sketch it in the first octant.
b) Maximize the volume of this rectangular box.
c) What are the dimensions & volume of this largest possible rectangular box?
**Step-by-Step Guide:**
1. **Volume Function Calculation:**
- Given the surface equation \( z = 10 - 2x - y^2 \), we need to express \( z \) as a function of \( x \) and \( y \).
- The volume \( V \) of the rectangular box in the first octant will be \( V = l \cdot w \cdot h \), where:
- \( l = x \) (length)
- \( w = y \) (width)
- \( h = z \) (height, which we get from the surface equation)
So, the volume function \( V \) in terms of \( x \) and \( y \) is:
\[
V(x, y) = x \cdot y \cdot (10 - 2x - y^2)
\]
2. **Maximization:**
- To maximize the volume, we need to take partial derivatives of \( V \) with respect to \( x \) and \( y \), and set them to zero to find critical points.
- Solve the system of equations obtained from the partial derivatives to find the values of \( x \) and \( y \) that maximize \( V \).
3. **Finding Dimensions and Volume:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae61a1d8-20ff-4791-b131-877cbba94656%2F771a37c4-f2d1-4a52-9c73-25bf3335b7a6%2Fss82ke_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the volume and the dimensions of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex on the surface \( z = 10 - 2x - y^2 \).
**Figures Description:**
1. The first figure is a 3D graph showing a surface in the first octant formed by the equation \( z = 10 - 2x - y^2 \). The surface appears as a curved shape coming downwards, where the height at each point is determined by the function of \( x \) and \( y \).
2. The second figure is a simple 3D coordinate system for sketching. It has labeled axes: \( x \), \( y \), and \( z \).
**Tasks:**
a) Find the volume function (in terms of \( x \) & \( y \)) for this rectangular box and sketch it in the first octant.
b) Maximize the volume of this rectangular box.
c) What are the dimensions & volume of this largest possible rectangular box?
**Step-by-Step Guide:**
1. **Volume Function Calculation:**
- Given the surface equation \( z = 10 - 2x - y^2 \), we need to express \( z \) as a function of \( x \) and \( y \).
- The volume \( V \) of the rectangular box in the first octant will be \( V = l \cdot w \cdot h \), where:
- \( l = x \) (length)
- \( w = y \) (width)
- \( h = z \) (height, which we get from the surface equation)
So, the volume function \( V \) in terms of \( x \) and \( y \) is:
\[
V(x, y) = x \cdot y \cdot (10 - 2x - y^2)
\]
2. **Maximization:**
- To maximize the volume, we need to take partial derivatives of \( V \) with respect to \( x \) and \( y \), and set them to zero to find critical points.
- Solve the system of equations obtained from the partial derivatives to find the values of \( x \) and \( y \) that maximize \( V \).
3. **Finding Dimensions and Volume:
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