A customer has asked you to design an open-top rectangular stainless steel vat. It is to have a square base and a volume of 32 ft³ , to be welded from quarter-inch plate, and to weigh no more than necessary. What dimensions do you recommend?

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Start by defining the variables: let x be the length of the side of the square base in feet, and h be the height of the vat in feet.

Since the vat has a square base and an open top, the volume V of the vat is given by the formula V = x^2 * h. We know the volume is 32 ft³, so set up the equation x^2 * h = 32.

To minimize the weight, which is proportional to the surface area of the material used, we need to minimize the surface area S of the vat. The surface area S is given by S = x^2 + 4xh, where x^2 is the area of the base and 4xh is the area of the four sides.

Use the volume equation to express h in terms of x: h = 32 / x^2. Substitute this expression for h into the surface area equation to get S in terms of x: S = x^2 + 4x(32 / x^2). Simplify this to S = x^2 + 128 / x.

To find the dimensions that minimize the surface area, take the derivative of S with respect to x, set it equal to zero, and solve for x. This will give you the optimal side length of the base. Then, use the volume equation to find the corresponding height h.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Volume of a Rectangular Prism

The volume of a rectangular prism is calculated by multiplying its length, width, and height. In this case, since the vat has a square base, the volume can be expressed as V = x²h, where x is the length of a side of the base and h is the height. Understanding this formula is crucial for determining the dimensions that will yield the required volume of 32 ft³.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function subject to certain constraints. In this problem, we need to minimize the surface area of the vat while maintaining a fixed volume. This requires setting up a function for surface area in terms of the dimensions and using techniques such as derivatives to find critical points.

Surface Area Calculation

The surface area of an open-top rectangular vat can be calculated using the formula A = x² + 4xh, where x is the side length of the base and h is the height. Since the vat is open at the top, this formula accounts for the base and the four sides. Understanding how to derive and manipulate this formula is essential for solving the optimization problem effectively.