Answer:
Explanation:
Minimizing surface area plays a large role in manufacturing. Companies try to use the smallest amount of resources possible to
package products in order to save money.
1. Calculate the surface area of the 2 figures shown (show all work), which both enclose the same volume.
2. Which container would you recommend a company use to package small candies?
3. Explain your reasoning.
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To compare the two figures and determine which container is more efficient in terms of surface area while enclosing the same volume, we can calculate the surface area of both shapes. The two common shapes used for this type of problem are a cube and a sphere. Let's assume both containers have the same volume.
Surface Area of a Cube:
The surface area of a cube is calculated using the formula A = 6s^2, where "s" is the length of a side of the cube. Since we're assuming both shapes enclose the same volume, we can use the volume formula for a cube to find the side length "s" first.
Let's assume the volume of the container is V. The volume of a cube is given by V = s^3. Therefore, s = V^(1/3).
Surface area of the cube = 6s^2 = 6(V^(2/3)).
Surface Area of a Sphere:
The surface area of a sphere is given by the formula A = 4πr^2, where "r" is the radius of the sphere. We can find the radius "r" using the volume formula for a sphere:
V = (4/3)πr^3, so r = (3V / 4π)^(1/3).
Surface area of the sphere = 4πr^2 = 4π((3V / 4π)^(2/3)).
Comparison and Recommendation:
To recommend which container to use for packaging small candies while minimizing surface area, we need to compare the surface area formulas for both the cube and the sphere.
Surface area of the cube: 6(V^(2/3)).
Surface area of the sphere: 4π((3V / 4π)^(2/3)).
The comparison depends on the specific volume of candies to be packaged. If you have a fixed volume (V) for the candies, you can calculate the surface area for both shapes and compare them directly. However, without knowing the actual volume (V), it's difficult to make a definitive recommendation. You would need to know the volume of the candies to make an informed decision about which container to use for packaging.
Keep in mind that in most cases, for a given volume, a sphere will have a slightly smaller surface area than a cube. This is due to the geometric properties of these shapes. However, practical considerations such as ease of manufacturing, transportation, and handling may also influence the choice of container.