Open Top Box Optimization Problem Sample

Problem Statement and Diagram

Given Variables

The objective is to determine the dimensions of an open-top box that maximize volume given a square base and a fixed surface area. Let x denote the length of the square base (in inches) and h the height of the box (in inches). The surface area comprises the square base and four rectangular sides. The surface area equation is S = x² + 4xh. The given constant is S = 108 square inches.

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Constraints

Physical dimensions require x > 0 and h > 0. Because 4xh = 108 - x², the inequality 108 - x² > 0 restricts the domain of the base dimension to 0 < x < √108.

Mathematical Formulation

Volume Equation

The volume V of a rectangular prism with a square base is the product of its base area and height: V = x²h.

Substitution

Isolating h from the surface area equation yields:

4xh = 108 - x²

h = (108 - x²) / (4x)

Substituting h into the volume equation expresses the objective function strictly in terms of x:

V(x) = x² [ (108 - x²) / (4x) ]

V(x) = x(108 - x²) / 4 = 27x - (1/4)x³

Optimization Process

First Derivative

Differentiating the objective function V(x) with respect to x identifies critical points:

V'(x) = 27 - (3/4)x²

Critical Points

Applying Fermat's theorem (Stewart, 2020, p. 331), local extrema occur where the first derivative equals zero:

27 - (3/4)x² = 0

(3/4)x² = 27

x² = 36

Because lengths are strictly positive, the valid critical point is x = 6 inches.

Verification and Conclusion

Second Derivative Test

Applying the Second Derivative Test (Thomas et al., 2018) confirms whether x = 6 maximizes volume. The second derivative is:

V''(x) = -(6/4)x = -(3/2)x

Evaluating at the critical point gives V''(6) = -(3/2)(6) = -9. Because V''(6) < 0, the function is concave down at x = 6, verifying a local maximum.

Final Dimensions

Substituting x = 6 into the expression for h yields the optimal height:

h = (108 - 6²) / (4 * 6) = (108 - 36) / 24 = 72 / 24 = 3 inches

The optimal dimensions are a 6-inch base length, 6-inch base width, and 3-inch height, producing a maximum volume of 108 cubic inches.

References

Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage Learning.

Thomas, G. B., Weir, M. D., & Hass, J. (2018). Thomas' Calculus (14th ed.). Pearson.