Xenia is ordering a cake for her friend’s wedding. She knows that there will be 110 guests at the wedding, and she is trying to decide how large a cake to order. She has been told that each guest should get a piece of cake that is 4 inches by 4 inches. She would like to buy a tiered, square three-layer cake and is trying to decide which sizes of cakes to order. Which combination of cakes should she get? The available sizes, in inches, are:
32 × 32
28 × 28
24 × 24
20 × 20
16 × 16
12 × 12
8 × 8
To determine which combination of cakes Xenia should get, we need to find the size of the cake that can accommodate 110 guests with each guest receiving a piece that is 4 inches by 4 inches.
First, let's calculate the area of a single piece of cake:
Area of a single piece of cake = Length x Width
= 4 inches x 4 inches
= 16 square inches
Next, we need to calculate the total area of cake needed for 110 guests:
Total area of cake needed = 110 guests x 16 square inches per guest
Now, let's calculate the total area of the cakes available in the given sizes:
For a 32 x 32 cake:
Total area = 32 inches x 32 inches = 1024 square inches
For a 28 x 28 cake:
Total area = 28 inches x 28 inches = 784 square inches
For a 24 x 24 cake:
Total area = 24 inches x 24 inches = 576 square inches
For a 20 x 20 cake:
Total area = 20 inches x 20 inches = 400 square inches
For a 16 x 16 cake:
Total area = 16 inches x 16 inches = 256 square inches
For a 12 x 12 cake:
Total area = 12 inches x 12 inches = 144 square inches
For a 8 x 8 cake:
Total area = 8 inches x 8 inches = 64 square inches
Now, let's compare the total area of cake needed (1760 square inches) with the total area of each cake available. We want to choose the combination of cake sizes that has the minimum number of tiers to avoid any leftover cake.
To minimize the number of tiers, we can start by using the largest available cake size (in this case, 32 x 32) and see if it meets or exceeds the total area of cake needed. If it does, we choose only that cake size. If not, we move to the next smaller cake size and repeat the process until we find the combination that satisfies the total area required.
In this case, the total area needed (1760 square inches) is greater than the area of a 32 x 32 cake (1024 square inches). Therefore, we need to include a smaller cake size.
Continuing the process, we find that a 32 x 32 cake combined with a 28 x 28 cake would provide enough cake to meet the total area needed:
Total area = 1024 square inches (32 x 32 cake) + 784 square inches (28 x 28 cake) = 1808 square inches
This combination exceeds the total area needed (1760 square inches). We cannot use only the 28 x 28 cake because its area alone is not enough.
Therefore, Xenia should get a combination of cakes: one 32 x 32 cake and one 28 x 28 cake to accommodate 110 guests and provide each guest with a 4 inches by 4 inches piece of cake.