7. An open-top box is to be made from a 30-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side. What is the side length a that should be cut out to get the maximum possible volume of the box? V = (length)(width)(height) 30 48 Give an equation, in terms of x for the quantity that you are trying to maximize.
7. An open-top box is to be made from a 30-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side. What is the side length a that should be cut out to get the maximum possible volume of the box? V = (length)(width)(height) 30 48 Give an equation, in terms of x for the quantity that you are trying to maximize.
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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![### Problem Statement:
7. An open-top box is to be made from a 30-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side. What is the side length \( x \) that should be cut out to get the maximum possible volume of the box?
\[ V = (\text{length})(\text{width})(\text{height}) \]
#### Diagram:
A diagram is provided below the problem statement. It shows a rectangle with dimensions 30 inches by 48 inches. There are squares cut out from each corner of the rectangle, each of side length \( x \).
- The dimensions of the rectangle are labeled: one side is labeled "30", and the other side is labeled "48".
- Each of the small squares cut out from the corners is labeled with the side length \( x \).
The resulting structure, once the squares are removed and the sides are folded up, will form an open-top box, and the dimensions of the box need to be determined to maximize its volume.
#### Equation:
Give an equation, in terms of \( x \), for the quantity that you are trying to maximize.
To write an equation for the volume \( V \) of the resulting box:
- The length of the box will be \( 48 - 2x \).
- The width of the box will be \( 30 - 2x \).
- The height of the box will be \( x \).
Therefore, the volume \( V \) of the box can be expressed as:
\[ V = (48 - 2x)(30 - 2x)(x) \]
This equation will allow you to determine the volume \( V \) as a function of \( x \), helping to find the value of \( x \) that maximizes the box's volume.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8759b09c-a99c-4616-917a-1eb84cf5071d%2F06dd1a8a-4729-49b1-8b44-fb34d1eb5424%2Fybz397b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
7. An open-top box is to be made from a 30-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side. What is the side length \( x \) that should be cut out to get the maximum possible volume of the box?
\[ V = (\text{length})(\text{width})(\text{height}) \]
#### Diagram:
A diagram is provided below the problem statement. It shows a rectangle with dimensions 30 inches by 48 inches. There are squares cut out from each corner of the rectangle, each of side length \( x \).
- The dimensions of the rectangle are labeled: one side is labeled "30", and the other side is labeled "48".
- Each of the small squares cut out from the corners is labeled with the side length \( x \).
The resulting structure, once the squares are removed and the sides are folded up, will form an open-top box, and the dimensions of the box need to be determined to maximize its volume.
#### Equation:
Give an equation, in terms of \( x \), for the quantity that you are trying to maximize.
To write an equation for the volume \( V \) of the resulting box:
- The length of the box will be \( 48 - 2x \).
- The width of the box will be \( 30 - 2x \).
- The height of the box will be \( x \).
Therefore, the volume \( V \) of the box can be expressed as:
\[ V = (48 - 2x)(30 - 2x)(x) \]
This equation will allow you to determine the volume \( V \) as a function of \( x \), helping to find the value of \( x \) that maximizes the box's volume.
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