To determine how much icing is needed to cover the cake, we need to calculate the surface area of the cake that will be covered in icing. We are given that the cake is a cylinder with a radius of 13 centimeters and a height of 15 centimeters. Additionally, the problem specifies that everything but the circular bottom of the cake is to be iced.
The steps are as follows:
1. Calculate the area of the circular top:
[tex]\[
\text{Top area} = \pi \times (\text{radius})^2
\][/tex]
Given the radius [tex]\( r = 13 \)[/tex] cm and [tex]\( \pi = 3.14 \)[/tex]:
[tex]\[
\text{Top area} = 3.14 \times 13^2 = 3.14 \times 169 = 530.66 \, \text{cm}^2
\][/tex]
2. Calculate the lateral surface area of the cylinder:
The lateral surface area is the area of the side of the cylinder, which can be obtained by the formula:
[tex]\[
\text{Lateral area} = 2 \times \pi \times \text{radius} \times \text{height}
\][/tex]
Given the height [tex]\( h = 15 \)[/tex] cm:
[tex]\[
\text{Lateral area} = 2 \times 3.14 \times 13 \times 15 = 2 \times 3.14 \times 195 = 2 \times 612.3 = 1224.6 \, \text{cm}^2
\][/tex]
3. Add the areas together:
Since we are icing the circular top and the lateral side of the cylinder:
[tex]\[
\text{Total icing area} = \text{Top area} + \text{Lateral area}
\][/tex]
Substituting the calculated values:
[tex]\[
\text{Total icing area} = 530.66 + 1224.6 = 1755.26 \, \text{cm}^2
\][/tex]
4. Round to the nearest square centimeter:
The total area rounded to the nearest square centimeter is:
[tex]\[
1755 \, \text{cm}^2
\][/tex]
So, the multiple-choice answer to how many square centimeters of icing is needed for one cake is:
[tex]\[
\boxed{1755 \, \text{cm}^2}
\][/tex]
