A box with a square base and open top must have a volume of 256000 cm³. Find the dimensions of the box that minimize the amount of material by completing the following steps. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. A(x) Next, find the derivative, A'(x). A'(x)= = The critical point is a Since there is only once critical point and since A'(x) is [Select an answer when x is less than the critical point but A'(x) is [Select an answer when x is greater than the critical point, the critical point corresponds to an absolute Select an answer When the length of a side of the square base is box is cm and when the height of the cm, the amount of material used to construct the box is minimized.
A box with a square base and open top must have a volume of 256000 cm³. Find the dimensions of the box that minimize the amount of material by completing the following steps. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. A(x) Next, find the derivative, A'(x). A'(x)= = The critical point is a Since there is only once critical point and since A'(x) is [Select an answer when x is less than the critical point but A'(x) is [Select an answer when x is greater than the critical point, the critical point corresponds to an absolute Select an answer When the length of a side of the square base is box is cm and when the height of the cm, the amount of material used to construct the box is minimized.
Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter63: Volumes Of Pyramids And Cones
Section: Chapter Questions
Problem 16A: A piece in the shape of a pyramid with a regular octagon (eight sided) base is machined from a solid...
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![## Optimization Problem: Minimizing Material for a Box
### Problem Statement
A box with a square base and open top must have a volume of \(256,000 \, \text{cm}^3\). Find the dimensions of the box that minimize the amount of material by completing the following steps.
### Steps to Solve the Problem
1. **Formula for Surface Area**
**Objective**: To find a formula for the surface area \(A\) of the box in terms of only \(x\), the length of one side of the square base.
\[
A(x) = \text{[ ]} \quad \text{(input formula here)}
\]
2. **Derivative Calculation**
**Objective**: Find the derivative, \(A'(x)\).
\[
A'(x) = \text{[ ]} \quad \text{(input derivative here)}
\]
3. **Finding the Critical Point**
**Objective**: Identify the critical point \(x\).
\[
\text{The critical point is } x = \text{[ ]}
\]
4. **Analysis of the Derivative**
Since there is only one critical point and since \(A'(x)\) is
\[
\text{(Select an answer)} \quad \text{when } x \text{ is less than the critical point but } A'(x) \text{ is } \text{(Select an answer)} \quad \text{when } x \text{ is greater than the critical point, the critical point corresponds to an absolute } \text{(Select an answer)}
\]
5. **Final Dimensions when Material is Minimized**
When the length of a side of the square base is
\[
\text{[ ]} \quad \text{cm and when the height of the box is } \text{[ ]} \quad \text{cm, the amount of material used to construct the box is minimized.}
\]
### Instructions:
Follow the outlined steps to determine the most efficient dimensions for the box, ensuring the surface area (hence, the material used) is minimized given the specified volume constraint. Make sure to compute the necessary derivatives and evaluate the critical points accurately to achieve the correct answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffdc195bb-9009-47db-866e-65b6514806e8%2F60fd529f-5340-4904-aaf7-090f79548a96%2Fk9o40vt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Optimization Problem: Minimizing Material for a Box
### Problem Statement
A box with a square base and open top must have a volume of \(256,000 \, \text{cm}^3\). Find the dimensions of the box that minimize the amount of material by completing the following steps.
### Steps to Solve the Problem
1. **Formula for Surface Area**
**Objective**: To find a formula for the surface area \(A\) of the box in terms of only \(x\), the length of one side of the square base.
\[
A(x) = \text{[ ]} \quad \text{(input formula here)}
\]
2. **Derivative Calculation**
**Objective**: Find the derivative, \(A'(x)\).
\[
A'(x) = \text{[ ]} \quad \text{(input derivative here)}
\]
3. **Finding the Critical Point**
**Objective**: Identify the critical point \(x\).
\[
\text{The critical point is } x = \text{[ ]}
\]
4. **Analysis of the Derivative**
Since there is only one critical point and since \(A'(x)\) is
\[
\text{(Select an answer)} \quad \text{when } x \text{ is less than the critical point but } A'(x) \text{ is } \text{(Select an answer)} \quad \text{when } x \text{ is greater than the critical point, the critical point corresponds to an absolute } \text{(Select an answer)}
\]
5. **Final Dimensions when Material is Minimized**
When the length of a side of the square base is
\[
\text{[ ]} \quad \text{cm and when the height of the box is } \text{[ ]} \quad \text{cm, the amount of material used to construct the box is minimized.}
\]
### Instructions:
Follow the outlined steps to determine the most efficient dimensions for the box, ensuring the surface area (hence, the material used) is minimized given the specified volume constraint. Make sure to compute the necessary derivatives and evaluate the critical points accurately to achieve the correct answer.
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