To find the volume of the cone with the given dimensions, we will use the formula for the volume of a cone:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Step-by-step solution:
1. Identify the given dimensions:
- The height ([tex]\( h \)[/tex]) of the cone is 10 yards.
- The diameter of the base is 12 yards.
2. Calculate the radius ([tex]\( r \)[/tex]) of the base:
Since the diameter is 12 yards,
[tex]\[ r = \frac{\text{diameter}}{2} = \frac{12}{2} = 6 \, \text{yards} \][/tex]
3. Substitute the known values into the volume formula:
[tex]\[ \pi \approx 3.14 \][/tex]
[tex]\[ h = 10 \, \text{yards} \][/tex]
[tex]\[ r = 6 \, \text{yards} \][/tex]
Plug these values into the formula:
[tex]\[ V = \frac{1}{3} \times 3.14 \times (6)^2 \times 10 \][/tex]
4. Calculate the volume:
- First, calculate [tex]\( (6)^2 \)[/tex]:
[tex]\[ 6^2 = 36 \][/tex]
- Next, multiply by [tex]\( \pi \)[/tex]:
[tex]\[ 3.14 \times 36 = 113.04 \][/tex]
- Then, multiply by the height:
[tex]\[ 113.04 \times 10 = 1130.4 \][/tex]
- Finally, multiply by [tex]\( \frac{1}{3} \)[/tex]:
[tex]\[ V = \frac{1}{3} \times 1130.4 \][/tex]
[tex]\[ V = 376.8 \][/tex]
Therefore, the volume of the cone is [tex]\( 376.8 \)[/tex] cubic yards.
