Final Answer:
The dimensions for the package that minimize production cost are a cylinder with a radius of approximately 3.18 centimeters and a height of 6.32 centimeters.
Step-by-step explanation:
To find the dimensions that minimize the production cost of the cup-of-soup package, we need to consider the surface area of the cylinder. The cost consists of the styrofoam for the sides and bottom, priced at 0.04 cents per square centimeter, and the paper for the top, priced at 0.07 cents per square centimeter. The surface area of the cylinder (A) is given by the formula A = 2πr² + 2πrh, where r is the radius and h is the height.
The volume of the cylinder is 200 cubic centimeters, given by V = πr²h. Using this volume equation, we can express the height (h) in terms of the radius (r): h = 200 / (πr²). Substituting this expression for height into the surface area formula, we get A = 2πr² + 2πr * (200 / πr²), simplifying to A = 2πr² + 400 / r.
To find the minimum cost, we need to minimize the surface area function. Calculating the derivative of the surface area with respect to the radius, setting it equal to zero, and solving for 'r' will yield the optimal radius. Upon solving, we find that the minimum cost occurs at a radius of approximately 3.18 centimeters. Substituting this value of radius back into the volume equation allows us to solve for the corresponding height, which is approximately 6.32 centimeters. These dimensions minimize the production cost of the cup-of-soup package.