# How do you find the dimensions that minimize the amount of cardboard used if a cardboard box without a lid is to have a volume of #8,788 (cm)^3#?

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To minimize the amount of cardboard used for a cardboard box without a lid with a volume of 8,788 (cm)^3, you need to find the dimensions that minimize the surface area of the box.

Let the length, width, and height of the box be represented by ( l ), ( w ), and ( h ) respectively. The volume of the box is given by the formula ( V = lwh ). Since there's no lid, the surface area ( A ) of the box is given by the formula ( A = 2lw + 2lh + wh ).

To find the dimensions that minimize ( A ), you can use calculus by differentiating ( A ) with respect to ( l ), ( w ), and ( h ), setting the derivatives equal to zero, and solving for each dimension.

However, since you are looking for a specific volume (( V = 8,788 ) ( cm^3 )), you can use the given volume to express one of the variables in terms of the other two using the formula ( V = lwh ). Then substitute this expression into the formula for the surface area ( A ). This will give you a function of two variables. You can then find the minimum of this function using calculus.

Solving this mathematically may involve some algebraic manipulation and calculus techniques, such as partial derivatives and critical points. Once you have the critical points, you can determine which point corresponds to the minimum surface area.

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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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