Final answer:
The dimensions of the lightest open-top can to hold a volume of 1000 cm³ are obtained by minimizing the surface area at a fixed volume. After using volume and surface area formulas for a cylinder, we find that the can with a radius to height ratio of 1:2 is most efficient. This corresponds to a radius of 5 cm and a height of 20 cm, which isn't listed but is the corrected form of option B.
Step-by-step explanation:
To determine the dimensions of the lightest open-top can that holds a volume of 1000 cm³, we must consider the material used for the sides and the bottom of the can. For the lightest can, we want to minimize the surface area, because the weight of the can is generally proportional to its surface area. The formula for the surface area S of an open-top cylindrical can is given by S = 2πrh + πr², where r is the radius and h is the height.
The volume V of a cylinder is given by V = πr²h. Since we know the volume needs to be 1000 cm³, we can set up an equation to solve for h in terms of r: 1000 = πr²h. By minimizing the surface area with a fixed volume, we find that the can with a ratio of height to radius equal to 2:1 is the most efficient, as it will have the least material used for a given volume. Therefore, the dimensions with the smallest surface area for a volume of 1000 cm³ are found with a radius of 5 cm and a height of 20 cm, which corresponds to a variation of option B with a correct height.