# Cavalieri's Principle and Volume of Composite Figures

A three-dimensional model of Lucia's apartment is shown. To cool the whole apartment, an air conditioner must provide 2.5 BTUs per cubic foot. How many BTUs are needed to cool Lucia's apartment?c

c. 20,400 BTUs

Which expression represents the volume, in cubic units, of the composite figure? ~4, 8, 6, 10

a. 1/3(8)(6)(6) + (8)(6)(4)

A cylindrical metal pipe has a diameter of 20 millimeters and a height of 21 millimeters. A cylindrical hole cut out of the center has a radius of 6 millimeters. ~20, 21, 6 Which expressions represent the volume of metal needed, in cubic millimeters, to make the pipe? Check all that apply.

a. 21π(10)^2 - 21π(6)^2 c. 2,100π - 756π

Which expression represents the volume, in cubic units, of the composite figure? ~20, 28

a. 4/3π(10^3) + π(10^2)(28)

What is the volume of the composite figure? ~4in, 7in, 3in, 12in

140 cubic inches

The composite figure is made up of two congruent rectangular pyramids joined at their bases. What is the total volume of the composite figure?

60 units^3

A frustum is formed when a plane parallel to a cone's base cuts off the upper portion as shown. ~6.4, 1.5 ~6.8, 3 fustrum What is the volume of the frustum? Express the answer in terms of π.

c. 34.8π units3

Cavalieri's principle states that if two solids of equal height have equal cross-sectional areas at every level parallel to the respective bases, then the two solids have equal volume. The two shaded solids both have a height of 2r units. At every level, the areas of the cross sections of both solids equal π(r2 - b2). Therefore, the formula for the volume of the sphere can be derived by writing an expression that represents the volume of

c. the solid between the two cones and the cylinder.

A rectangular prism with a square base is cut from the center of a triangular prism. Each side of the square base measures 1.7 units. ~5.7, 1.7, 5, 4 What is the volume of the shaded solid? Round to the nearest tenth of a cubic unit.

45.4 units ^3

A volleyball of radius 13 inches fits exactly in a cylinder with the same radius. ~26in cylinder Inside the container there is empty space not taken up by the volleyball. What is the volume of that empty space? Use 3.14 for π and round the answer to the nearest cubic inch.

b. 4,599 in.^3

The large sphere has a diameter of 20 feet. ~4ft Which expression represents the volume, in cubic units, of the shaded part of the sphere?

b. 4/3π(10^3) - 4/3π(4^3)

What is the volume of the composite figure? Use 3.14 for π and round the answer to the nearest tenth of a cubic unit. ~11, 20, 12

c. 1,582.6 in.^3

A frustum is formed when a plane parallel to a cone's base cuts off the upper portion as shown. ~8, 3.5 ~11, 7.5 fustrum Which expression represents the volume, in cubic units, of the frustum?

c. 1/3π(7.5^2)(19) - 1/3π(3.5^2)(8)

This composite figure is made of two identical pyramids attached at their bases. Each pyramid has a height of 2 units. ~2, 5, 0.25 Which expression represents the volume, in cubic units, of the composite figure?

c. 2(1/3(5)(0.25)(2))