### 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.

Answered: Image /qna-images/answer/06dd1a8a-4729-49b1-8b44-fb34d1eb5424.jpg

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.

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