Question
A castle-themed birthday cake is made from a 10-inch square pan with a height of 2 inches. The cake also has cones on each corner of the square that have a radius of 114 inches and a slant height of 314 inches. Each cone’s base rests completely atop the cake’s surface. There are 3 layers of the square base of the castle, meaning only the lateral sides and the top face are part of the surface area. Determine the number of square inches of fondant needed to cover the cake.(1 point) The baker will need square inches of fondant.
Answers
Answered by
GPT-4o mini
To determine the amount of fondant needed to cover the castle-themed birthday cake, we need to calculate the surface area of the cake.
### Step 1: Calculate the surface area of the cake.
The cake is a square prism with dimensions:
- Base: 10 inches x 10 inches
- Height: 2 inches
#### Lateral Area of the Cake
The lateral area (side area) of a rectangular prism (here, a square prism) can be calculated using the formula:
\[
\text{Lateral Area} = \text{Perimeter of the base} \times \text{Height}
\]
The perimeter \(P\) of the base of the cake is:
\[
P = 4 \times \text{side length} = 4 \times 10 \text{ in} = 40 \text{ in}
\]
Now, multiply by the height:
\[
\text{Lateral Area} = 40 \text{ in} \times 2 \text{ in} = 80 \text{ in}^2
\]
#### Area of the Top Face
The area of the top face of the cake is simply the area of the square base:
\[
\text{Area of Top Face} = \text{side length}^2 = 10 \text{ in} \times 10 \text{ in} = 100 \text{ in}^2
\]
### Step 2: Total Surface Area of the Cake
Add the lateral area and the top face area together:
\[
\text{Total Surface Area of Cake} = \text{Lateral Area} + \text{Area of Top Face} = 80 \text{ in}^2 + 100 \text{ in}^2 = 180 \text{ in}^2
\]
### Step 3: Calculate the surface area of the cones.
Each cone has:
- Radius \( r = 1.14 \) inches
- Slant height \( s = 3.14 \) inches
#### Lateral Area of a Cone
The lateral surface area \(A\) of a cone can be calculated using:
\[
A = \pi r s
\]
Substituting the values:
\[
A = \pi \times 1.14 \times 3.14 \approx 3.14159 \times 1.14 \times 3.14 \approx 11.28 \text{ in}^2
\]
Since there are 4 cones, we multiply this area by 4:
\[
\text{Total Lateral Area of Cones} = 4 \times 11.28 \text{ in}^2 \approx 45.12 \text{ in}^2
\]
### Step 4: Combine Areas
Now, we need to add the total area of the cake and the total area of the cones:
\[
\text{Total Surface Area} = \text{Total Surface Area of Cake} + \text{Total Lateral Area of Cones}
\]
\[
= 180 \text{ in}^2 + 45.12 \text{ in}^2 = 225.12 \text{ in}^2
\]
### Final Calculation
The final amount of fondant needed to cover the whole cake is approximately:
\[
\text{Total Fondant needed} \approx 225.12 \text{ in}^2
\]
Thus, the baker will need **approximately 225 square inches of fondant** to cover the cake.
### Step 1: Calculate the surface area of the cake.
The cake is a square prism with dimensions:
- Base: 10 inches x 10 inches
- Height: 2 inches
#### Lateral Area of the Cake
The lateral area (side area) of a rectangular prism (here, a square prism) can be calculated using the formula:
\[
\text{Lateral Area} = \text{Perimeter of the base} \times \text{Height}
\]
The perimeter \(P\) of the base of the cake is:
\[
P = 4 \times \text{side length} = 4 \times 10 \text{ in} = 40 \text{ in}
\]
Now, multiply by the height:
\[
\text{Lateral Area} = 40 \text{ in} \times 2 \text{ in} = 80 \text{ in}^2
\]
#### Area of the Top Face
The area of the top face of the cake is simply the area of the square base:
\[
\text{Area of Top Face} = \text{side length}^2 = 10 \text{ in} \times 10 \text{ in} = 100 \text{ in}^2
\]
### Step 2: Total Surface Area of the Cake
Add the lateral area and the top face area together:
\[
\text{Total Surface Area of Cake} = \text{Lateral Area} + \text{Area of Top Face} = 80 \text{ in}^2 + 100 \text{ in}^2 = 180 \text{ in}^2
\]
### Step 3: Calculate the surface area of the cones.
Each cone has:
- Radius \( r = 1.14 \) inches
- Slant height \( s = 3.14 \) inches
#### Lateral Area of a Cone
The lateral surface area \(A\) of a cone can be calculated using:
\[
A = \pi r s
\]
Substituting the values:
\[
A = \pi \times 1.14 \times 3.14 \approx 3.14159 \times 1.14 \times 3.14 \approx 11.28 \text{ in}^2
\]
Since there are 4 cones, we multiply this area by 4:
\[
\text{Total Lateral Area of Cones} = 4 \times 11.28 \text{ in}^2 \approx 45.12 \text{ in}^2
\]
### Step 4: Combine Areas
Now, we need to add the total area of the cake and the total area of the cones:
\[
\text{Total Surface Area} = \text{Total Surface Area of Cake} + \text{Total Lateral Area of Cones}
\]
\[
= 180 \text{ in}^2 + 45.12 \text{ in}^2 = 225.12 \text{ in}^2
\]
### Final Calculation
The final amount of fondant needed to cover the whole cake is approximately:
\[
\text{Total Fondant needed} \approx 225.12 \text{ in}^2
\]
Thus, the baker will need **approximately 225 square inches of fondant** to cover the cake.