Final answer:
To minimize the construction materials necessary for a cylindrical container holding 828 cubic centimeters of fluid, we can optimize the surface area. By expressing the height in terms of the radius, we can find the critical points of the surface area formula and determine if they minimize the surface area.
Step-by-step explanation:
To determine the radius of the cylindrical container that minimizes the construction materials necessary, we need to optimize the surface area. Since the volume of the cylinder must be 828 cubic centimeters, we can use the volume formula for a cylinder, V = πr²h, to express the height in terms of the radius. Substituting this expression for the height into the surface area formula, A = 2πrh + 2πr², we can find the derivative with respect to r and set it equal to zero to find the critical points. Finally, we can evaluate the second derivative to make sure that the critical point minimizes the surface area.