Modern Design magazine is having a contest to design the best ice cube in one of three shapes—a cube, a sphere, or a cylinder. Use the following guidelines to make your choice and complete your contest entry.
1. Find and simplify the volume-to-surface-area ratio for each of the three possible ice cube shapes.
Create a visual model to show your ice cube design, including the dimensions you would choose and the volume and surface area of that cube.
Your visual model and the explanation of your choice of ice cube shape and size will be submitted as your portfolio assessment.
To determine the best ice cube design, we'll first calculate the volume-to-surface-area ratio for each of the three shapes. The higher the ratio, the better the design, as it means the ice will have a larger volume and take longer to melt.
1. Calculate the Volume-to-Surface-Area Ratio
a) Cube
Volume (V): V = a³ (where a is the side length)
Surface Area (SA): SA = 6a²
Volume-to-Surface-Area ratio for a cube:
V/SA = a³/(6a²) = a/6
b) Sphere
Volume (V): V = (4/3)πr³ (where r is the radius)
Surface Area (SA): SA = 4πr²
Volume-to-Surface-Area ratio for a sphere:
V/SA = [(4/3)πr³] / (4πr²) = r/3
c) Cylinder
Volume (V): V = πr²h (where r is the radius and h is the height)
Surface Area (SA): SA = (2πr²) + (2πrh)
Volume-to-Surface-Area ratio for a cylinder:
V/SA = [πr²h] / [(2πr²) + (2πrh)]
We can see that the ratio depends on the dimensions chosen for the cylinder, unlike the cube and sphere ratios, which only depend on their respective side lengths and radii.
2. Selecting the Best Design and Dimensions
As we compare the ratio formulas, we can see that the sphere has the highest volume-to-surface-area ratio, which is desirable for the ice cube because it'll last longer. We will now choose an appropriate size for the sphere.
For a sphere's radius of 1 inch:
- Volume: V = (4/3)π(1)³ = 4.19 cubic inches (approximately)
- Surface Area: SA = 4π(1)² = 12.57 square inches (approximately)
- Volume-to-Surface-Area Ratio: r/3 = 1/3 = 0.33
3. Creating the Visual Model
For the final ice cube design, we'll create a visual model of the sphere and provide dimensions and calculations:
Ice Cube Shape: Sphere
Radius: 1 inch
Volume: 4.19 cubic inches (approximately)
Surface Area: 12.57 square inches (approximately)
Volume-to-Surface-Area Ratio: 0.33
Our visual model can be a 3D rendering or simple drawing of a sphere, labeled with its radius and the calculated volume and surface area.
In conclusion, we have designed an ice cube in the shape of a sphere with a radius of 1 inch. This ice cube will have a higher volume-to-surface-area ratio than the cube and cylinder options, which means it'll last longer and provide more cooling capacity throughout its life. Our portfolio assessment will include the visual model, design choice explanation, and supporting calculations.