To find the volume of a right circular cylinder with the given parameters, we need to follow these steps:
1. Convert the height to a decimal:
[tex]\( 6 \frac{1}{4} = 6 + 0.25 = 6.25 \)[/tex]
2. Calculate the diameter:
The diameter is 3.5 times the height.
[tex]\[
\text{Diameter} = 3.5 \times 6.25 = 21.875
\][/tex]
3. Determine the radius:
The radius is half of the diameter.
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{21.875}{2} = 10.9375
\][/tex]
4. Use the formula for the volume of a cylinder:
The volume [tex]\( V \)[/tex] of a cylinder is given by [tex]\( V = \pi r^2 h \)[/tex].
Substituting [tex]\( \pi = 3.14 \)[/tex], [tex]\( r = 10.9375 \)[/tex], and [tex]\( h = 6.25 \)[/tex]:
[tex]\[
V = 3.14 \times (10.9375^2) \times 6.25
\][/tex]
5. Calculate the radius squared:
[tex]\[
10.9375^2 = 119.62890625
\][/tex]
6. Calculate the volume:
[tex]\[
V = 3.14 \times 119.62890625 \times 6.25
\][/tex]
[tex]\[
V = 3.14 \times 747.6806640625
\][/tex]
[tex]\[
V = 2347.6752867953125
\][/tex]
7. Round to the nearest hundredth:
[tex]\[
\text{Volume} \approx 2347.68
\][/tex]
Therefore, the volume of the cylinder is [tex]\( \boxed{2347.68} \)[/tex].
