The complete question in the attached figure
we know that
Tower B ( lower left)
a) Square Pyramid
V = 1/3 lwh
V = (1/3)(3)(3)(3)
V = (1/3)(3)(9)
V = (1/3)(27)
V = 9 cubic units
b) Rectangular Prism
V = lwh
V = (3)(50)(3)
V = (3)(150)
V = 450 cubic units
tower B volume
Va + Vb
450 + 9
459 cubic units
Tower E (lower right)
a) Cone
V = 1/3 pi r^2 h
V = (1/3)(3.14)(3^2)(3)
V = (1/3)(3.14)(9)(3)
V = (1/3)(3.14)(27)
V = (1/3)(84.78)
V = 28.26 cubic units
b) Cylinder
V = pi r^2 h
V = (3.14)(3^2)(50)
V = (3.14)(9)(50)
V = (3.14)(450)
V = 1,413 cubic units
Tower E Volume
Va + Vb
28.26 + 1,413
1,441.26 cubic units
Tower A (upper left)
a) Hemisphere
Since it is a hemisphere, divide the formula of sphere by 2.
V = (4/3)pi r^3 all over by 2
V = (4/3)(3.14)(3^3) all over by 2
V = 113.04 / 2
V = 56.52 cubic units
b) Cylinder
V = pi r^2 h
V = (3.14)(3^2)(50)
V = (3.14)(9)(50)
V = (3.14)(450)
V = 1,413 cubic units
3rd Tower Volume
Va + Vb
56.52 + 1,413
1,469.52 cubic units
Tower D (upper right)
a) Triangular pyramid
V = 1/3(1/2 bh)(H)
where b is base of the triangular base
h is the height of the triangular base
H is the altitude of the pyramid
Since H is unknown, bisect the triangular base then use Pythagorean theorem to find H.
a^2 + b^2 = c^2
let a be the base of the right triangle
b be the H or missing side of the right triangle
c be the hypotenuse of the triangle
(1.5^2) + (b^2) = 3^2
2.25 + b^2 = 9
b^2 = 9 - 2.25
b^2 = 6.75
b = 2.6 units
H = 2.6 units
Substitute:
V = (1/3)[(1/2)(3)(2.6)](3)
V = (1/3)[3.9](3)
V = (1/3)(11.7)
V = 3.9 cubic units
b) Triangular Prism
V = (bh/2) H
V = [(1.5)(2.6)/2](50)
V = (3.9/2)(50)
V = (1.95)(50)
V = 97.5 cubic units
4th Tower Volume
Va + Vb
3.9 + 97.5
101.4 cubic units
Main Castle
V = lwh
V = (100)(50)(30)
V = (100)(1500)
V = 150,000 cubic units
Total Volume
V1 + V2 + V3 + V4 + V5
459 + 1,441.26 + 1,469.52 + 101.4 + 150,000 ------> 153,471.18 cubic units
Therefore,
the answer is
the volume of the castle and the towers is 153,471.18 cubic units
