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Big Ideas Math Geometry, 2014

BI

Big Ideas Math Geometry, 2014 View details

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Exercise 3 Page 661

Find the volume of the cone-shaped sections and the volume of the cylinder-shaped section separately.

128 π ≈ 402.1 m^3

Practice makes perfect

We want to find the volume of a composite solid in the shape of an oblique cylinder with cones attached on each end.

Oblique cylinder with a cone attached on each end

We will first find the volume of the cones and then the volume of the cylinder. Then we can add the results to find the volume of the solid. Let's do it!

Volume of the Cones

Let's begin by finding the volume of the cone at the top.

Oblique cylinder with a cone attached on each end. Top cone highlighted

The volume of a cone is one third the product of the area of the base and the height. V=1/3 Bh Our cone has a circular base, B, and its area is the product of π times the radius squared. V=1/3 Bh ⇒ V = 1/3 π r^2 h Let's substitute the corresponding values into the formula and find its volume.

V = 1/3 π r^2 h

r= 4, h= 3

V = 1/3 π ( 4)^2 ( 3)

▼

Simplify right-hand side

Calculate power

V = 1/3 π (16) (3)

MoveRightFacToNumOne

1/b* a = a/b

V = π (16) (3)/3

CancelCommonFac

Cancel out common factors

V = π (16)

CommutativePropMult

Commutative Property of Multiplication

V = 16π

The cone has a volume of 16 πm^3. Next we want to find the volume of the cone at the bottom of the solid.

Oblique cylinder with a cone attached on each end. Bottom cone highlighted

Since this cone has the same dimensions as the other cone, we know that it too must have a volume of 16 πm^3.

Volume of the Cylinder

The midsection of the solid is in the shape of an oblique cylinder.

Oblique cylinder with a cone attached on each end. Cylinder highlighted

The volume of a cylinder is the product of the area of the base and the height. The base is a circle and its area is the product of π and the radius squared. V=Bh ⇒ V = π r^2 h Our cylinder has a radius of 4 meters and a height of 6 meters. Let's find its volume.

V = π r^2 h

r= 4, h= 6

V = π ( 4)^2 ( 6)

▼

Simplify right-hand side

Calculate power

V = π (16) ( 6)

Multiply

V = π (96)

CommutativePropMult

Commutative Property of Multiplication

V = 96 π

Volume of the Solid

To find the volume of the composite solid we will use the Volume Addition Postulate.

The volume of a solid is the sum of the volumes of all its non-overlapping parts.

The solid has three non-overlapping parts — two cones and one cylinder. Let's add their volumes. V_(solid)= V_(cone)+ V_(cone)+ V_(cylinder) ⇓ V_(solid)= 16π+ 16π+ 96π = 128 π The solid has the volume 128 π ≈ 402.1 m^3.

Subchapter links

Exercises p.661