Question

1. you are baking a tiered cake that you want to use as a model for a large party. the bottom tier will use a 10 inch (diameter) pan, and the top will use a 6 inch (diameter) pan. both cake pans have the same depth. the cake looks like this.

what type of solid are these cake tiers?
what is the volume of the bottom tier?
what is the volume of the top tier?
what is the total volume of the cake?

Explanation:

Step1: Determine the formula for the volume of a cylinder

The volume formula for a cylinder is \(V=\pi r^{2}h\).

Step2: Find the radius of the bottom - tier cylinder

The diameter of the bottom - tier (10 - inch pan) is \(d = 10\) inches. So the radius \(r_{1}=\frac{d}{2}=\frac{10}{2}=5\) inches. The depth \(h = 4\) inches.
Substitute into the formula: \(V_{1}=\pi\times(5)^{2}\times4=\pi\times25\times4 = 100\pi\approx314\) cubic inches.

Step3: Find the radius of the top - tier cylinder

The diameter of the top - tier (6 - inch pan) is \(d = 6\) inches. So the radius \(r_{2}=\frac{d}{2}=\frac{6}{2}=3\) inches. The depth \(h = 4\) inches.
Substitute into the formula: \(V_{2}=\pi\times(3)^{2}\times4=\pi\times9\times4=36\pi\approx113.04\) cubic inches.

Step4: Calculate the total volume

\(V = V_{1}+V_{2}\). Substitute \(V_{1}\approx314\) and \(V_{2}\approx113.04\)
\(V\approx314 + 113.04=427.04\) cubic inches.

Answer:

The type of solid is a cylinder. The volume of the bottom tier is approximately \(314\) cubic inches. The volume of the top tier is approximately \(113.04\) cubic inches. The total volume of the cake is approximately \(427.04\) cubic inches.