Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 14 in. by 9 in. by cutting congruent squares from the corners and folding up the sides. Then find the volume. (Round to the nearest 100th.)
Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 14 in. by 9 in. by cutting congruent squares from the corners and folding up the sides. Then find the volume. (Round to the nearest 100th.)
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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Question
![**Problem Statement:**
Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard measuring 14 inches by 9 inches. This is done by cutting congruent squares from the corners and folding up the sides. Calculate the volume and round to the nearest hundredth.
**Explanation:**
To form the open rectangular box, we start with a rectangular sheet of cardboard with dimensions 14 inches by 9 inches. The process involves the following steps:
1. **Cutting Squares:**
- Congruent squares are cut from each corner of the rectangular sheet. The side length of each square is denoted as \(x\).
2. **Folding the Sides:**
- After cutting the squares, fold up the remaining flaps to form the sides of the open box.
3. **Dimensions of the Box:**
- The resulting box will have dimensions:
- Length: \(14 - 2x\)
- Width: \(9 - 2x\)
- Height: \(x\)
4. **Volume Calculation:**
- The volume \(V\) of the box can be expressed as the product of its dimensions:
\[ V = x(14 - 2x)(9 - 2x) \]
5. **Finding Maximum Volume:**
- To find the value of \(x\) that maximizes the volume, you will typically derive the volume function and find its critical points. Solve for \(x\) considering \(0 < x < 4.5\), since the dimensions must remain positive.
6. **Round Results:**
- Once the maximum volume is determined, round the result to the nearest hundredth for precision.
By following these steps, you can determine the optimal dimensions and volume for the open box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F939f19e0-6626-477a-ba7a-4282434e880a%2Fdc18a2a7-4862-436d-b3f3-0958aad03e54%2Fgcaet6ac_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard measuring 14 inches by 9 inches. This is done by cutting congruent squares from the corners and folding up the sides. Calculate the volume and round to the nearest hundredth.
**Explanation:**
To form the open rectangular box, we start with a rectangular sheet of cardboard with dimensions 14 inches by 9 inches. The process involves the following steps:
1. **Cutting Squares:**
- Congruent squares are cut from each corner of the rectangular sheet. The side length of each square is denoted as \(x\).
2. **Folding the Sides:**
- After cutting the squares, fold up the remaining flaps to form the sides of the open box.
3. **Dimensions of the Box:**
- The resulting box will have dimensions:
- Length: \(14 - 2x\)
- Width: \(9 - 2x\)
- Height: \(x\)
4. **Volume Calculation:**
- The volume \(V\) of the box can be expressed as the product of its dimensions:
\[ V = x(14 - 2x)(9 - 2x) \]
5. **Finding Maximum Volume:**
- To find the value of \(x\) that maximizes the volume, you will typically derive the volume function and find its critical points. Solve for \(x\) considering \(0 < x < 4.5\), since the dimensions must remain positive.
6. **Round Results:**
- Once the maximum volume is determined, round the result to the nearest hundredth for precision.
By following these steps, you can determine the optimal dimensions and volume for the open box.
Expert Solution
Step 1: Given Information:
Given that a sheet of cardboard 14 in. by 9 in. by cutting congruent squares from the corners and folding up the sides.
To find the dimensions of the open rectangular box of maximum volume that can be
made from a given sheet of cardboard.
Step by step
Solved in 4 steps with 14 images
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