A box with a square base and open top must have a volume of 157216 cm3. We wish to find the dimensions of the box that minimize the amount of material used. 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)=
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 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.

Step-by-step explanation:

Given a box with a square base and an open top which must have a volume of 157216 cubic centimetre. We want to minimize the amount of material used.

Step 1:

Let the side length of the base =x

Let the height of the box =h

Since the box has a square base

Volume

Surface Area of the box = Base Area + Area of 4 sides

Step 2: Find the derivative of A(x)

Step 3: Set A'(x)=0 and solve for x

Step 4: Verify that x=68 is a minimum value

We use the second derivative test

Since the second derivative is positive at x=68, then it is a minimum point.

Recall:

Therefore, the dimensions that minimizes the box surface area are:

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step-by-step explanation:

basically 2/5.

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step-by-step explanation:

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