To find the volume of a right circular cone, we use the formula:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
where:
- [tex]\( V \)[/tex] is the volume,
- [tex]\( r \)[/tex] is the radius of the base,
- [tex]\( h \)[/tex] is the height of the cone,
- [tex]\( \pi \)[/tex] is a mathematical constant approximately equal to 3.14159.
Given:
- The height ([tex]\( h \)[/tex]) of the cone is 3.5 feet,
- The diameter of the base is 11.9 feet.
First, we need to calculate the radius ([tex]\( r \)[/tex]) of the base. The radius is half of the diameter:
[tex]\[ r = \frac{\text{diameter}}{2} = \frac{11.9}{2} = 5.95 \text{ feet} \][/tex]
Next, we substitute the values of [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into the volume formula:
[tex]\[ V = \frac{1}{3} \pi (5.95)^2 (3.5) \][/tex]
Now, we calculate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = (5.95)^2 = 35.4025 \][/tex]
Therefore,
[tex]\[ V = \frac{1}{3} \pi (35.4025) (3.5) \][/tex]
Now, calculate [tex]\( 35.4025 \times 3.5 \)[/tex]:
[tex]\[ 35.4025 \times 3.5 = 123.90875 \][/tex]
Next, we multiply by [tex]\( \pi \)[/tex]:
[tex]\[ V \approx \frac{1}{3} \times 3.14159 \times 123.90875 \][/tex]
Next, calculate [tex]\( 3.14159 \times 123.90875 \)[/tex]:
[tex]\[ 3.14159 \times 123.90875 \approx 389.6847 \][/tex]
Finally, multiply by [tex]\( \frac{1}{3} \)[/tex]:
[tex]\[ V \approx \frac{389.6847}{3} \approx 129.9 \][/tex]
Thus, the volume of the cone, rounded to the nearest tenth, is approximately [tex]\( 129.9 \)[/tex] cubic feet.
