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Stage 5.1-3

7.09 Further composite solids

Worksheet
Volumes of truncated pyramids and cones
1

Consider the diagram of a small square pyramid with a height of x\text{ cm} and a base side length of y\text{ cm}. A large pyramid has dimensions double that of the small pyramid.

a

State the dimensions of the large pyramid.

b

Find an expression for the volume of the large pyramid.

c

How many times can the volume of the small pyramid go into the volume of the large pyramid?

d

If the small pyramid has a volume of 59\text{ cm}^3, find the volume of the large pyramid.

2

A small square pyramid has a height of x\text{ m} and a base side length of y\text{ m}. A large pyramid has dimensions triple that of the small pyramid.

a

State the dimensions of the large pyramid.

b

Find an expression for the volume of the large pyramid.

c

How many times can the volume of the small pyramid go into the volume of the large pyramid?

d

If the small pyramid has a volume of 46\text{ m}^3, find the volume of the large pyramid.

3

Consider a pyramid which has a height of p\text{ m} and a base area of q\text{ m}^2.

a

If its side lengths are scaled up by a factor of four, state the new height and base area.

b

Write an expression for the volume of the pyramid from part (a) after the side lengths are scaled up by a factor of four.

c

When a pyramid has side lengths scaled up by a factor of four, by what factor does the volume increase?

d

A model pyramid has a height of 16\text{ cm} and a base area of 21\text{ cm}^2. If it is a 1:40 scale model of an actual pyramid, find the volume of the actual pyramid.

4

A pyramid has a volume of 648\,000\text{ m}^3. If a model is made in the ratio 1:60, determine the volume of the model.

5

Frank has a small cone with a height of x\text{ cm} and a radius of y\text{ cm} and a large cone which has dimensions double that of the small cone.

a

State the dimensions of the large cone.

b

Find an expression for the exact volume of the large cone.

c

How many times can the volume of the small cone go into the volume of the large cone?

d

If the small cone has a volume of 63\text{ cm}^3, find the volume of the large cone.

6

Caitlin has a small cone with a height of x\text{ cm} and a radius of y\text{ cm} and a large cone which has dimensions triple that of the small cone.

a

State the dimensions of the large cone.

b

Find an expression for the exact volume of the large cone.

c

How many times can the volume of the small cone go into the volume of the large cone?

d

If the small cone has a volume of 28\text{ cm}^3, find the volume of the large cone.

7

Rochelle has a red traffic cone and an orange traffic cone. The red traffic cone has dimensions four times that of the orange cone.

a

How many times greater is the volume of the red traffic cone compared to the orange one?

b

If the volume of the red cone is 20\,480\text{ cm}^3, find the volume of the orange cone.

8

Homer has a blue traffic cone and a yellow traffic cone. The blue traffic cone has dimensions four times that of the yellow cone.

If the volume of the yellow cone is 353\text{ cm}^3, find the volume of the blue cone.

9

Laura makes a frustum by cutting off a square pyramid halfway from the top, as shown in the diagram below:

a

Find an expression for the exact volume of the original pyramid.

b

Find an expression for the volume of the pyramid section that was cut from the original pyramid.

c

What fraction of the original pyramid did Laura cut off?

d

If the original pyramid had a volume of 608\text{ cm}^3, find the exact volume of Laura's frustum.

10

Zane makes a frustum by cutting off a square pyramid a third of the way from the top, as shown in the diagram below:

a

Find an expression for the exact volume of the original pyramid.

b

Find an expression for the volume of the pyramid section that was cut from the original pyramid.

c

What fraction of the original pyramid did Zane cut off?

d

If the original pyramid had a volume of 1161\text{ cm}^3, find the exact volume of Zane's frustum.

11

Lucy makes a truncated cone by cutting off a smaller cone halfway from the top, as shown in the diagram below:

a

Find an expression for the exact volume of the original cone.

b

Find an expression for the exact volume of the cone section that was cut from the original cone.

c

What fraction of the original cone did Lucy cut off?

d

If the original cone had a volume of 184\text{ cm}^3, find the exact volume of Lucy's truncated cone.

12

James makes a truncated cone by cutting a cone a third of the way from the top, as shown in the diagram below:

a

Find an expression for the exact volume of the original cone.

b

Find an expression for the exact volume of the cone section that was cut from the original cone.

c

What fraction of the original cone did James cut off?

d

If the original cone had a volume of 567\text{ cm }^3, find the exact volume of James's truncated cone.

13

Consider the diagram of a solid constructed from two identical frustums. Each frustum was made by removing the top third of the original square pyramid.

Find the volume of the solid, correct to two decimal places.

14

An hourglass-like solid is constructed from two identical truncated cones. Each truncated cone is half the height of the original cone.

Find the volume of the solid, correct to two decimal places.

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Outcomes

MA5.3-13MG

applies formulas to find the surface areas of right pyramids, right cones, spheres and related composite solids

MA5.3-14MG

applies formulas to find the volumes of right pyramids, right cones, spheres and related composite solids

MA5.3-16MG

proves triangles are similar, and uses formal geometric reasoning to establish properties of triangles and quadrilaterals

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