. A box with a top is constructed from a piece of 200-cm by 400-cm cardboard. The shaded regions are cut out and the flaps are folded to make a box as in the diagram below. Write a polynomial in x for the volume of the box. What are the dimensions that give the maximum volume of the box? 400cm bottom top 200cm
. A box with a top is constructed from a piece of 200-cm by 400-cm cardboard. The shaded regions are cut out and the flaps are folded to make a box as in the diagram below. Write a polynomial in x for the volume of the box. What are the dimensions that give the maximum volume of the box? 400cm bottom top 200cm
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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please help, please answer all subparts if any, thank you.
![### Maximizing the Volume of a Box Constructed from a Cardboard Sheet
**Problem Statement:**
A box with a top is constructed from a piece of 200-cm by 400-cm cardboard. The shaded regions are cut out, and the flaps are folded to make a box, as illustrated in the diagram below. The task is to write a polynomial in \(x\) for the volume of the box and determine the dimensions that give the maximum volume.
**Diagram Explanation:**
- The cardboard piece has dimensions of 400 cm in length and 200 cm in width.
- Square sections with side length \(x\) are cut out from each corner of the cardboard sheet.
- The remaining flaps are then folded up to form the sides of the box, creating a box with a top.
The yellow region represents the bottom and top of the box once the flaps are folded.
**Volume Calculation:**
1. **Dimensions of the Box:**
- **Length**: After cutting out squares from each end, the new length is \(400 - 2x\).
- **Width**: Similarly, the new width is \(200 - 2x\).
- **Height**: The height of the box is determined by the side length of the cut squares, which is \(x\).
2. **Volume Formula:**
The volume \(V\) of the box can be expressed as:
\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]
Substituting the dimensions:
\[
V = (400 - 2x) \times (200 - 2x) \times x
\]
Simplifying, the volume polynomial is:
\[
V(x) = x (400 - 2x) (200 - 2x)
\]
Further expanding:
\[
V(x) = x (400 \times 200 - 400 \times 2x - 200 \times 2x + 4x^2)
\]
\[
V(x) = x (80000 - 1200x + 4x^2)
\]
\[
V(x) = 80000x - 1200x^2 + 4x^3
\]
The polynomial representing the volume](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06eccc32-203a-4225-9aef-79cfa83d100a%2Fbafd18b9-5844-4614-8ec4-e2c82c82f413%2Fxqjh4s_processed.png&w=3840&q=75)
Transcribed Image Text:### Maximizing the Volume of a Box Constructed from a Cardboard Sheet
**Problem Statement:**
A box with a top is constructed from a piece of 200-cm by 400-cm cardboard. The shaded regions are cut out, and the flaps are folded to make a box, as illustrated in the diagram below. The task is to write a polynomial in \(x\) for the volume of the box and determine the dimensions that give the maximum volume.
**Diagram Explanation:**
- The cardboard piece has dimensions of 400 cm in length and 200 cm in width.
- Square sections with side length \(x\) are cut out from each corner of the cardboard sheet.
- The remaining flaps are then folded up to form the sides of the box, creating a box with a top.
The yellow region represents the bottom and top of the box once the flaps are folded.
**Volume Calculation:**
1. **Dimensions of the Box:**
- **Length**: After cutting out squares from each end, the new length is \(400 - 2x\).
- **Width**: Similarly, the new width is \(200 - 2x\).
- **Height**: The height of the box is determined by the side length of the cut squares, which is \(x\).
2. **Volume Formula:**
The volume \(V\) of the box can be expressed as:
\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]
Substituting the dimensions:
\[
V = (400 - 2x) \times (200 - 2x) \times x
\]
Simplifying, the volume polynomial is:
\[
V(x) = x (400 - 2x) (200 - 2x)
\]
Further expanding:
\[
V(x) = x (400 \times 200 - 400 \times 2x - 200 \times 2x + 4x^2)
\]
\[
V(x) = x (80000 - 1200x + 4x^2)
\]
\[
V(x) = 80000x - 1200x^2 + 4x^3
\]
The polynomial representing the volume
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