Ok so what we have is a large cylinder with a hollow cylindrical tube running thru the center. The volume of the shaded region should be the volume of the entire large cylinder (C) minus the volume of the small cylindrical tube (c): C - c
The volume of any cylinder is the area of the top circle (pi×r^2) times the height (h) of the cylinder, where r=radius of the circle. So: h×pi×r^2
The heights are the same for C and c, so we can call that h in both, which = 30ft. Now C's radius (R) = 1/2 of diameter, which is 18ft. So R = 9ft
c's radius (r) is 1/2 of its diameter, which is 6ft. So r = 3ft. Now we can set it up:
C - c = (h×pi×R^2) - (h×pi×r^2)
*since the h and pi in both formulae are multiplied by the other variable, we can factor that out:
h×pi(R^2) - h×pi(r^2) = h×pi(R^2-r^2)
h×pi = 30ft.×3.14 = 94.2ft. -->
94.2ft(9ft^2-3ft^2) = 94.2ft(81-9ft^2)
= 94.2ft(72ft^2) = 6782.4 ft.^3