To determine how much cement is needed to fill each cone-shaped top, we first need to use the formula for the volume of a cone. The formula for the volume [tex]\( V \)[/tex] of a cone is:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the base of the cone,
- [tex]\( h \)[/tex] is the height of the cone, and
- [tex]\( \pi \)[/tex] is approximately 3.14.
Given:
- The height of the cone [tex]\( h \)[/tex] is 12 cm,
- The base radius of the cone [tex]\( r \)[/tex] is 5 cm.
Let's plug these values into the volume formula:
[tex]\[ V = \frac{1}{3} \times 3.14 \times (5)^2 \times 12 \][/tex]
First, we need to calculate the area of the base (which is a circle):
[tex]\[ \text{Base Area} = \pi r^2 = 3.14 \times (5)^2 = 3.14 \times 25 = 78.5 \, \text{cm}^2 \][/tex]
Now, multiply the base area by the height and then by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ V = \frac{1}{3} \times 78.5 \times 12 \][/tex]
[tex]\[ V = \frac{1}{3} \times 942 \][/tex]
[tex]\[ V = 314 \, \text{cm}^3 \][/tex]
Therefore, the amount of cement needed for each cone-shaped top is:
[tex]\[ \boxed{314 \, \text{cm}^3} \][/tex]
