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.

Question

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.

Answer 1

To start this project, we need to find and simplify the volume-to-surface-area ratio for each of the three possible ice cube shapes: a cube, a sphere, and a cylinder.

1. Cube:

Let's denote the side length of the cube as "s".

Volume (V) of a cube = s³
Surface area (SA) of a cube = 6s²

Now, we calculate the volume-to-surface-area ratio for the cube:

V/SA = (s³) / (6s²)
Simplifying this, we get (s³) / (6s²) = s/6

2. Sphere:

Let the radius of the sphere be "r".

Volume (V) of a sphere = (4/3)πr³
Surface area (SA) of a sphere = 4πr²

Now, we calculate the volume-to-surface-area ratio for the sphere:

V/SA = ((4/3)πr³) / (4πr²)
Simplifying this, we get ((4/3)πr³) / (4πr²) = r/3

3. Cylinder:

Let the radius of the base of the cylinder be "r" and the height of the cylinder be "h".

Volume (V) of a cylinder = πr²h
Surface area (SA) of a cylinder = 2πr(h + r)

Now, we calculate the volume-to-surface-area ratio for the cylinder:

V/SA = (πr²h) / (2πr(h + r))
Simplifying this, we get (πr²h) / (2πr(h + r)) = h / (2(h + r))

Now that we have the volume-to-surface-area ratios for each shape, let's choose a shape and size for our ice cube design.

The higher the volume-to-surface-area ratio, the slower the ice will melt, as there will be a smaller surface area exposed to heat. Comparing the simplifications we obtained, we can observe that the sphere has the highest ratio when compared to the cube and the cylinder (when we compare the coefficients since r and h are arbitrary). This means that a spherical ice cube will melt slower, keeping our drink cold for a longer period of time.

Let's choose a sphere with a radius of 2cm.

Volume (V) of our chosen sphere = (4/3)π(2)³ = 32π/3 ≈ 33.51 cm³
Surface area (SA) of our chosen sphere = 4π(2)² = 16π ≈ 50.27 cm²

Now, create a visual model of the sphere with its radius, volume, and surface area labeled.

Lastly, we need to write an explanation of our choice:

For our ice cube design, we chose a sphere with a radius of 2 cm because it has a higher volume-to-surface-area ratio, which means that it will melt slower than other shapes, keeping our drink cold for a longer period of time. The volume of this sphere is approximately 33.51 cm³, and the surface area is approximately 50.27 cm².