6 Open-box Problem. An open-box (top open) is made from a rectangular material of dimensions $ a = 15 $ inches by $ b = 12 $ inches by cutting a square of side $ x $ at each corner and turning up the sides (see the figure). Determine the value of $ x $ that results in a box with the maximum volume.**Diagram Explanation:**The diagram shows a rectangle with length $ a = 15 $ inches and width $ b = 12 $ inches. Smaller squares with side $ x $ are marked in each of the four corners. The area between these squares is highlighted in yellow, representing the base of the box that is formed when the sides are folded upwards.Following the steps to solve the problem. Check Show Answer only after you have tried hard.1. **Express the volume $ V $ as a function of $ x $:** $ V = $ [Input box] 2. **Determine the domain of the function $ V $ of $ x $ (in interval form):** [Input box] 3. **Expand the function $ V $ for easier differentiation:** $ V = $ [Input box] **Mathematical Exercise: Finding Key Values in a Function**1. **Derivative of the Function** (4) Find the derivative of the function $ V $: $ V' = $ [Input box]2. **Critical Points** (5) Find the critical point(s) in the domain of $ V $: [Input box]3. **Function Values at Endpoints** (6) The value of $ V $ at the left endpoint is [Input box] (7) The value of $ V $ at the right endpoint is [Input box]4. **Maximum Volume** (8) The maximum volume is $ V = $ [Input box]5. **Conclusion** (9) Answer the original question. The value of $ x $ that maximizes the volume is: [Input box]This exercise guides students through the process of differentiating a function, identifying critical points, and evaluating the function at endpoints to find the maximum value of $ V $.

Name: 6 Open-box Problem. An open-box (top open) is made from a rectangular
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Description: 6 Open-box Problem. An open-box (top open) is made from a rectangular material of dimensions $ a = 15 $ inches by $ b = 12 $ inches by cutting a square of side $ x $ at each corner and turning up the sides (see the figure). Determine the value

We are given that an open-box (top open) is made from a rectangular material of dimensions a = 15 inches by b = 12 inches by cutting a square of side x at each corner and turning up the sides (see the figure). We need to determine the value of x that results in a box the maximum volume. The figure of the problem looks like the following. Form the figure, it can be observed that the volume V(x)=a-2x(b-2x)(x)=>V(x)=(15-2x)(12-2x)(x)=>V(x)=4x3-54x2+180x. ------(1) Note: As per the question, we will be answering (6). To answer (6), we first need to find the values of x. So, V'(x)=12x2-108x+180. For the extremum point, V'(x)=0=>12x2-108x+180=0=>x=--108±-1082-4· 12· 1802· 12=>x=9+212, 9-212=>x=6.79128, 2.20871 (approx.) The left endpoint of x is 2.20871. Answer: So, the volume at the left endpoint isV(2.20871)=177.23408 (approx.). The right endpoint of x is 6.7912. So, the volume at the left endpoint isV(6.7912)=-15.23408 (approx.) . Thus, the volume (maximum) is V(2.20871)=177.23408 (approx.). Verification: Now, V''(x)=24x-108. For x = 6.7912, V''(x)=54.99072 >0, and for x=2.20871, V''(x)=-54.99096<0. Thus, the volume V(x) is maximum when x = 2.20871. And thus the volume (maximum) is V(2.20871)=177.23408 (approx.).

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Advanced Math

Open-box Problem. An open-box (top open) is made from a rectangular material of dimensions a = 15 inches by b = 12 inches by cutting a square of side x at each corner and turning up the sides (see the figure). Determine the value of x that results in a box the maximum volume. 13 12 10 구 9 10 11 12 13 14 15 16 Following the steps to solve the problem. Check Show Answer only after you have tried hard. (1) Express the volume V as a function of x: V : (2) Determine the domain of the function V of x (in interval form): (3) Expand the function V for easier differentiation: V =

Open-box Problem. An open-box (top open) is made from a rectangular material of dimensions a = 15 inches by b = 12 inches by cutting a square of side x at each corner and turning up the sides (see the figure). Determine the value of x that results in a box the maximum volume. 13 12 10 구 9 10 11 12 13 14 15 16 Following the steps to solve the problem. Check Show Answer only after you have tried hard. (1) Express the volume V as a function of x: V : (2) Determine the domain of the function V of x (in interval form): (3) Expand the function V for easier differentiation: V =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Open-box Problem. An open-box (top open) is made from a rectangular material of dimensions $ a = 15 $ inches by $ b = 12 $ inches by cutting a square of side $ x $ at each corner and turning up the sides (see the figure). Determine the value of $ x $ that results in a box with the maximum volume.

**Diagram Explanation:**
The diagram shows a rectangle with length $ a = 15 $ inches and width $ b = 12 $ inches. Smaller squares with side $ x $ are marked in each of the four corners. The area between these squares is highlighted in yellow, representing the base of the box that is formed when the sides are folded upwards.

Following the steps to solve the problem. Check Show Answer only after you have tried hard.

1. **Express the volume $ V $ as a function of $ x $:** $ V = $ [Input box]

2. **Determine the domain of the function $ V $ of $ x $ (in interval form):** [Input box]

3. **Expand the function $ V $ for easier differentiation:** $ V = $ [Input box]

**Mathematical Exercise: Finding Key Values in a Function**

1. **Derivative of the Function**

(4) Find the derivative of the function $ V $: $ V' = $
[Input box]

2. **Critical Points**

(5) Find the critical point(s) in the domain of $ V $:
[Input box]

3. **Function Values at Endpoints**

(6) The value of $ V $ at the left endpoint is
[Input box]

(7) The value of $ V $ at the right endpoint is
[Input box]

4. **Maximum Volume**

(8) The maximum volume is $ V = $
[Input box]

5. **Conclusion**

(9) Answer the original question. The value of $ x $ that maximizes the volume is:
[Input box]

This exercise guides students through the process of differentiating a function, identifying critical points, and evaluating the function at endpoints to find the maximum value of $ V $.

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