py your Tould as much as possible. A(x)= Next, find the derivative, A'(x). A'(x): Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x² .] A'(x) = 0 when x We next have to make sure that this value of a gives a minimum value for the surface area. Let's use the

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Question
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2
**Mathematics: Optimization Problems**

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**Simplify your formula as much as possible.**

\[ A(x) = \]

---

**Next, find the derivative, \( A'(x) \).**

\[ A'(x) = \]

---

**Now, calculate when the derivative equals zero, that is, when \( A'(x) = 0 \).** \[ \text{[Hint: multiply both sides by } x^2 \text{]} \]

\[ A'(x) = 0 \text{ when } x = \]

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**We next have to make sure that this value of \( x \) gives a minimum value for the surface area. Let’s use the second derivative test. Find \( A''(x) \).**

\[ A''(x) = \]

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**Evaluate \( A''(x) \) at the \( x \)-value you gave above.**

\[ \]

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**Note:** Since your last answer is positive, this means that the graph of \( A(x) \) is concave up around that value, so the zero of \( A'(x) \) must indicate a local minimum for \( A(x) \). (Your boss is happy now.)

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### Explanation of Diagram/Formula Usage

This worksheet is designed to guide students through the process of finding critical points by solving the derivative equations (first and second derivatives) to confirm minimum values in optimization problems. There are no graphs or diagrams provided in this image, but if there were, details such as axes labels, plotted points, slopes, curves, and any relevant features would be described to help in understanding the visual data representation. When a graph is provided in similar exercises, it usually features the function \( A(x) \) and its derivatives \( A'(x) \) and \( A''(x) \), illustrating how the curve behaves around critical points.

---

If you have any questions about the content or need further assistance with the mathematical concepts, please reach out to us!

**Happy Learning!**
Transcribed Image Text:**Mathematics: Optimization Problems** --- **Simplify your formula as much as possible.** \[ A(x) = \] --- **Next, find the derivative, \( A'(x) \).** \[ A'(x) = \] --- **Now, calculate when the derivative equals zero, that is, when \( A'(x) = 0 \).** \[ \text{[Hint: multiply both sides by } x^2 \text{]} \] \[ A'(x) = 0 \text{ when } x = \] --- **We next have to make sure that this value of \( x \) gives a minimum value for the surface area. Let’s use the second derivative test. Find \( A''(x) \).** \[ A''(x) = \] --- **Evaluate \( A''(x) \) at the \( x \)-value you gave above.** \[ \] --- **Note:** Since your last answer is positive, this means that the graph of \( A(x) \) is concave up around that value, so the zero of \( A'(x) \) must indicate a local minimum for \( A(x) \). (Your boss is happy now.) --- ### Explanation of Diagram/Formula Usage This worksheet is designed to guide students through the process of finding critical points by solving the derivative equations (first and second derivatives) to confirm minimum values in optimization problems. There are no graphs or diagrams provided in this image, but if there were, details such as axes labels, plotted points, slopes, curves, and any relevant features would be described to help in understanding the visual data representation. When a graph is provided in similar exercises, it usually features the function \( A(x) \) and its derivatives \( A'(x) \) and \( A''(x) \), illustrating how the curve behaves around critical points. --- If you have any questions about the content or need further assistance with the mathematical concepts, please reach out to us! **Happy Learning!**
**Minimizing Material in a Box with Square Base and Open Top**

A box with a square base and open top must have a volume of 296352 cm³. We wish to find the dimensions of the box that minimize the amount of material used. 

**Step 1: Find a Formula for Surface Area**

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. 
[Hint: use the volume formula to express the height of the box in terms of \( x \).]
Simplify your formula as much as possible.

\[ A(x) = \]

**Step 2: Find the Derivative**

Next, find the derivative, \( A'(x) \).

\[ A'(x) = \]

**Step 3: Set Derivative to Zero**

Now, calculate when the derivative equals zero, that is, when \( A'(x) = 0 \). 
[Hint: multiply both sides by \( x^2 \).]

\[ A'(x) = 0 \text{ when } x = \]

**Step 4: Verify Minimum Value with Second Derivative Test**

We next have to make sure that this value of \( x \) gives a minimum value for the surface area. Let’s use the second derivative test. Find \( A''(x) \).

\[ A''(x) = \]

**Step 5: Evaluate Second Derivative**

Evaluate \( A''(x) \) at the \( x \)-value you gave above.

\[ \]
Transcribed Image Text:**Minimizing Material in a Box with Square Base and Open Top** A box with a square base and open top must have a volume of 296352 cm³. We wish to find the dimensions of the box that minimize the amount of material used. **Step 1: Find a Formula for Surface Area** 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. [Hint: use the volume formula to express the height of the box in terms of \( x \).] Simplify your formula as much as possible. \[ A(x) = \] **Step 2: Find the Derivative** Next, find the derivative, \( A'(x) \). \[ A'(x) = \] **Step 3: Set Derivative to Zero** Now, calculate when the derivative equals zero, that is, when \( A'(x) = 0 \). [Hint: multiply both sides by \( x^2 \).] \[ A'(x) = 0 \text{ when } x = \] **Step 4: Verify Minimum Value with Second Derivative Test** We next have to make sure that this value of \( x \) gives a minimum value for the surface area. Let’s use the second derivative test. Find \( A''(x) \). \[ A''(x) = \] **Step 5: Evaluate Second Derivative** Evaluate \( A''(x) \) at the \( x \)-value you gave above. \[ \]
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