To determine the total amount of fondant needed to cover the cake, we need to calculate the surface area of the cake and the four cones.
Step 1: Calculate the surface area of the square cake
The cake is a square prism with:
- Length of each side of the base = 10 inches
- Height = 2 inches
The surface area \( A_{\text{cake}} \) consists of:
- The lateral sides (4 sides) of the square prism.
- The top face (since there are 3 layers we also consider only 1 top face for fondant).
Lateral areas of the sides: Each side has an area of: \[ \text{Area of one side} = \text{height} \times \text{base length} = 2 , \text{inches} \times 10 , \text{inches} = 20 , \text{square inches} \] Thus, the total lateral area from the four sides is: \[ \text{Lateral area} = 4 \times 20 , \text{square inches} = 80 , \text{square inches} \]
Top face: The area of the top face (one layer) is: \[ \text{Area of the top face} = \text{base length} \times \text{base length} = 10 , \text{inches} \times 10 , \text{inches} = 100 , \text{square inches} \]
Total surface area of the cake: \[ A_{\text{cake}} = \text{Lateral area} + \text{Top face area} = 80 + 100 = 180 , \text{square inches} \]
Step 2: Calculate the surface area of the cones
Each cone has:
- Radius \( r = 1 , \text{inch} \)
- Slant height \( l = 3 , \text{inches} \)
The surface area \( A_{\text{cone}} \) of one cone (lateral area only, since the base is not covered) is given by: \[ A_{\text{cone}} = \pi r l = \pi \cdot 1 \cdot 3 = 3\pi , \text{square inches} \]
Since there are 4 cones, the total surface area for all cones is: \[ A_{\text{cones}} = 4 \cdot 3\pi = 12\pi , \text{square inches} \]
Step 3: Total surface area requiring fondant
Thus, the total area for fondant will be: \[ \text{Total surface area} = A_{\text{cake}} + A_{\text{cones}} = 180 + 12\pi \]
Using \( \pi \approx 3.14 \) for calculations: \[ A_{\text{cones}} \approx 12 \cdot 3.14 = 37.68 , \text{square inches} \] \[ \text{Total surface area} \approx 180 + 37.68 = 217.68 , \text{square inches} \]
We can round this to: \[ \text{Total surface area} \approx 218 , \text{square inches} \]
Thus, the baker will need approximately 218 square inches of fondant.