Final answer:
To minimize the material used for a cylindrical container that holds 828 cubic centimeters, one should find the optimal radius by solving the equation V = πr²h for h, substitute into the surface area formula A = 2πrh + 2πr², and then take the derivative with respect to r, set it to zero, and solve for r.
Step-by-step explanation:
To determine the radius of a cylinder so that the required construction materials are minimized while still holding 828 cubic centimeters of fluid, one must use principles of calculus to optimize the surface area with the given volume. We start by using the formula for the volume of a cylinder, V = πr²h, where V is the volume, r is the radius, and h is the height. Given that we want the cylinder to hold 828 cm³, we can plug this volume into our equation to find a relationship between the radius and the height.
Next, to minimize the material used, we need to minimize the surface area of the cylinder, which includes both the sides and the two ends. The formula for the surface area of a cylinder is A = 2πrh + 2πr². Since we have an expression for h based on the volume and radius, we can substitute that into the formula for the surface area, ending up with an equation that only contains r. We then take the derivative of that expression with respect to r, set the derivative equal to zero, and solve to find the optimal value of r.
The optimal radius obtained will minimize the surface area and, hence, the material needed for construction. It's important to note that when we take the derivative, we are looking for critical points which would indicate either a minimum or maximum. After finding the critical points, we should test them to ensure we are looking at the minimum by using either the second derivative test or by comparison of values.