# An open-top box with a square base has a surface area of 1200 square inches. How do you find the largest possible volume of the box?

I tried this:

So the Volume will be:

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To find the largest possible volume of the box, you need to maximize the volume function with respect to the dimensions of the box. Let x be the length of one side of the square base and h be the height of the box.

Given that the surface area of the box is 1200 square inches, the surface area equation is: (1200 = x^2 + 4xh)

To express the volume of the box in terms of a single variable, we rewrite the height in terms of x: (h = \frac{1200 - x^2}{4x})

The volume function V(x) is: (V(x) = x^2 \cdot \frac{1200 - x^2}{4x})

To find the critical points of the volume function, differentiate V(x) with respect to x and set it equal to zero: (V'(x) = \frac{dV}{dx} = 0)

After finding the critical points, evaluate V(x) at those points and at the endpoints of the domain [0, ∞) to determine the maximum volume.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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