5 **Finding the Dimensions of a Box with Minimum Material**A box with a square base and an open top must have a volume of \(219488 \, \text{cm}^3\). We wish to find the dimensions of the box that minimize the amount of material used.**Step 1: Surface Area Formula**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: Derivative of Surface Area**Next, find the derivative, \( A'(x) \).\[ A'(x) = \]**Step 3: Finding Critical Points**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 Surface Area**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: Evaluating the Second Derivative**Evaluate \( A''(x) \) at the \( x \)-value you gave above.

Name: 5 **Finding the Dimensions of a Box with Minimum Material**A box with
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: 5 **Finding the Dimensions of a Box with Minimum Material**A box with a square base and an open top must have a volume of \(219488 \, \text{cm}^3\). We wish to find the dimensions of the box that minimize the amount of material used.**Step 1: Surface