We need to find the volume of the cylinder, then the cone and subtract to get the excavated volume.
[tex]\begin{gathered} \text{Vol of Cylinder =}\pi r^2h \\ \text{Vol of cone = }\frac{1}{3}\pi r^2h \\ Excavated\text{ volume = }\pi r^2h\text{ - }\frac{1}{3}\pi r^2h\text{ = }\pi r^2h(1-\frac{1}{3})=\frac{2}{3}\pi r^2h \end{gathered}[/tex]
We then substitute the values and find our solution.
where height, h = 8ft
radius = diameter / 2 = 3/2= 1.5 ft
[tex]\begin{gathered} \text{Vol of Cylinder =}\pi\times1.5^2\text{ }\times8=56.55\text{ cu ft} \\ \text{Vol of Cone =}\frac{1}{3}\times\pi\times1.5^2\text{ }\times8=18.85\text{ cu ft} \\ \text{Excavated volume =}\frac{2}{3}\times\pi\times1.5^2\text{ }\times8=37.7\text{ cu ft} \end{gathered}[/tex]
V cylinder = 56.55 cu ft
V Cone = 18.85 cu ft
V remaining = 37.7 cu ft
