A box with a square base and open top must have a volume of 296352 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. 2 A(x) = x² + Next, find the derivative, A'(x). 1185408 A'(x) = 2x- 1185408 X - The critical point is a esc Since there is only once critical point and since A'(x) is [negative critical point but A'(x) is positive point corresponds to an absolute minimum 84 When the length of a side of the square base is 84 the box is ✓when z is less than the when x is greater than the critical point, the critical cm and when the height of cm, the amount of material used to construct the box is minimized. Enter an integer or decimal number [more.. Question Help: Video 1 Video 2 Written Example 1 Submit Question F1 30% 80 F3 888 FA FS MacBook A
A box with a square base and open top must have a volume of 296352 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. 2 A(x) = x² + Next, find the derivative, A'(x). 1185408 A'(x) = 2x- 1185408 X - The critical point is a esc Since there is only once critical point and since A'(x) is [negative critical point but A'(x) is positive point corresponds to an absolute minimum 84 When the length of a side of the square base is 84 the box is ✓when z is less than the when x is greater than the critical point, the critical cm and when the height of cm, the amount of material used to construct the box is minimized. Enter an integer or decimal number [more.. Question Help: Video 1 Video 2 Written Example 1 Submit Question F1 30% 80 F3 888 FA FS MacBook A
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:**
A box with a square base and open top must have a volume of 296352 cm³. Find the dimensions of the box that minimize the amount of material by completing the following steps.
**Step 1: Find a formula for the surface area of the box in terms of only \( x \), the length of one side of the square base.**
\[ A(x) = x^2 + \frac{1185408}{x} \]
**Step 2: Find the derivative, \( A'(x) \).**
\[ A'(x) = 2x - \frac{1185408}{x^2} \]
**Step 3: Determine the critical point.**
The critical point is \( x = 84 \).
**Step 4: Analyze the behavior of the derivative to determine whether the critical point corresponds to a minimum.**
Since there is only one critical point and since \( A'(x) \) is **negative** when \( x \) is less than the critical point but \( A'(x) \) is **positive** when \( x \) is greater than the critical point, the critical point corresponds to an absolute **minimum**.
**Step 5: Calculate the dimensions when the length of a side of the square base is 84 cm.**
When the length of a side of the square base is \( 84 \) cm and when the height of the box is \( \_\_\_\_\_\_\_\_\_\_\_\_ \) cm, the amount of material used to construct the box is minimized.
**User Input Fields:**
- Enter height of the box in cm: [Input field for user to enter value]
- Submit button
**Resources for Assistance:**
1. Video Help 1
2. Video Help 2
3. Written Example 1
**Submission:**
- Submit Question (Button for submitting the inputted solution)
This structure guides learners through the problem-solving process step-by-step, incorporating calculus concepts such as derivative and critical point analysis to find the optimal dimensions for minimizing the material used for the box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffdc195bb-9009-47db-866e-65b6514806e8%2F2adeb965-1a88-40a3-ad6d-41b4e80f9a07%2Fbq0q0ys_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A box with a square base and open top must have a volume of 296352 cm³. Find the dimensions of the box that minimize the amount of material by completing the following steps.
**Step 1: Find a formula for the surface area of the box in terms of only \( x \), the length of one side of the square base.**
\[ A(x) = x^2 + \frac{1185408}{x} \]
**Step 2: Find the derivative, \( A'(x) \).**
\[ A'(x) = 2x - \frac{1185408}{x^2} \]
**Step 3: Determine the critical point.**
The critical point is \( x = 84 \).
**Step 4: Analyze the behavior of the derivative to determine whether the critical point corresponds to a minimum.**
Since there is only one critical point and since \( A'(x) \) is **negative** when \( x \) is less than the critical point but \( A'(x) \) is **positive** when \( x \) is greater than the critical point, the critical point corresponds to an absolute **minimum**.
**Step 5: Calculate the dimensions when the length of a side of the square base is 84 cm.**
When the length of a side of the square base is \( 84 \) cm and when the height of the box is \( \_\_\_\_\_\_\_\_\_\_\_\_ \) cm, the amount of material used to construct the box is minimized.
**User Input Fields:**
- Enter height of the box in cm: [Input field for user to enter value]
- Submit button
**Resources for Assistance:**
1. Video Help 1
2. Video Help 2
3. Written Example 1
**Submission:**
- Submit Question (Button for submitting the inputted solution)
This structure guides learners through the problem-solving process step-by-step, incorporating calculus concepts such as derivative and critical point analysis to find the optimal dimensions for minimizing the material used for the box.
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