A cylindrical gas container is closed at the top and open at...

A cylindrical gas container is closed at the top and open at the botton. If the iron plate of the top is $5 / 4$ times as thick as the plate forming the cylindrical sides, the ratio of the radius to the height of the cylinder using minimum material for the same capacity is (A) $3 ; 4$ (B) $5 ; 6$ (C) $4: 5$ (D) None of these

A cylindrical gas container is closed at the top and open at the botton. If the iron plate of the top is $5 / 4$ times as thick as the plate forming the cylindrical sides, the ratio of the radius to the height of the cylinder using minimum material for the same capacity is
(A) $3 ; 4$
(B) $5 ; 6$
(C) $4: 5$
(D) None of these

Key Concepts

Weighted Surface Area Calculation

In many practical design problems, different parts of an object may contribute unequally to the overall cost or material usage. When these parts have different thicknesses, densities, or costs, their areas are 'weighted' accordingly in the total material calculation. This adjustment is vital in finding the optimal design when the cost factors differ between components of the structure.

Cylindrical Geometry

Understanding the geometry of cylinders is crucial in problems involving cylindrical containers. This includes knowledge of how to compute the volume, lateral surface area, and the areas of circular ends. In optimization problems, these formulas help create equations that relate different dimensions (such as radius and height) to one another under a volume constraint.

Constrained Optimization

This concept revolves around optimizing (minimizing or maximizing) a function subject to certain restrictions or constraints. In the context of design problems, such as minimizing the amount of material used while achieving a prescribed volume, one typically forms a function representing material usage and then applies a constraint based on the geometric requirement, using techniques like setting derivatives to zero or the method of Lagrange multipliers.

Calculus-based Optimization

Calculus-based optimization involves using differential calculus to find the local minima or maxima of a function. By differentiating the cost or material function with respect to the geometric parameters and setting these derivatives to zero, one can locate the optimum dimensions. This approach is essential when designing objects with minimal material use while satisfying a fixed volume constraint.

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