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9.03 Composite solids

Worksheet
Volume of composite solids
1

Find the volume of the following composite solids. Round your answers to two decimal places where necessary.

a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
2

A triangular tunnel is made through a rectangular prism as shown in the figure. Find the volume of the solid formed.

3

The larger prism has had two identical holes carved out of it, each of which is a rectangular prism. Find the volume of the remaining solid, correct to two decimal places. All measurements are in metres.

Surface areas of composite solids
4

Find the surface area of the following composite solids. Round your answers to two decimal places where necessary.

a
b
c
d
e
f
5

The solid shown is constructed by cutting out a quarter of a sphere from a cube. Find its surface area if the side length is 14.2 \text{ cm} and the radius of the sphere is half the side length.

Applications
6

A wedding cake with three tiers is shown. The layers have radii of 51\text{ cm}, 55\text{ cm} and 59\text{ cm}. If each layer is 20\text{ cm} high, calculate the total volume of the cake in cubic metres.

Round your answer to two decimal places.

7

A swimming pool has the shape of a trapezoidal prism as shown in the diagram:

a

Find the volume of the pool in \text{m}^{3}.

b

If the pool is three-quarters full, what is the volume of the non filled space of the pool?

c

If the distance of the water level to the top of the pool is h \text{ m} when it is is three-quarters full, find h.

8

Consider the hollow pipe shown:

a

Find the external surface area of the curved surface. Round your answer to two decimal places.

b

Find the total surface area of the two end pieces. Round your answer to two decimal places.

c

Find the internal surface area. Round your answer to two decimal places.

d

Hence, find the total surface area. Round your answer to two decimal places.

9

The lid of this treasure chest is found to be exactly one half of a cylindrical barrel. Find the surface area of the chest, correct to two decimal places.

10

The given diagram shows the design for a marquee (tent). The roof of the marquee has a height of 3 \text{ m}. The material for the marquee costs \$44 per square metre.

a

Find the area of the front of the marquee.

b

Find the surface area of one of the side walls (not including the roof).

c

Find the surface area of the entire roof.

d

Find the total surface area. Do NOT include the floor.

e

Find the total cost of the marquee material.

11

A company manufactures nuts shaped like regular hexagonal prisms, with cylindrical bolt holes cut out of the centre, as shown below:

a

The total surface area of a nut before the bolt hole is drilled is 14.7\text{ cm}^2. Find the surface area of a single nut after the bolt hole is drilled out, including the inside surface area of the hole. Round your answer to one decimal place.

b

Each nut that is manufactured requires a zinc coating to prevent corrosion. If 1 \text{ kg} of zinc is enough to coat a surface area of 1\text{ m}^2, how many nuts can be coated with 1 \text{ kg} of zinc? Round your answer to the nearest whole number.

12

The swimming pool shown is composed of a trapezoidal prism joined to a half cylinder:

a

Find the volume of the pool in cubic metres. Round your answer to three decimal places.

b

How many litres of water can fit in the pool? Round your answer to the nearest litre.

c

If the pool is filled to a height 10\text{ cm} below the top, how many litres of water are in the pool? Round your answer to the nearest litre.

d

After construction works at a neighbouring property, a crack opens in the bottom of the pool and water begins to leak from the pool. It is observed that the height of the surface of the water in the pool is decreasing by 7\text{ cm} each week. Find the amount of water that is leaking out each week, to the nearest litre.

e

Assuming that water continues to leak at this rate, find how many whole weeks it will take to empty the pool.

13

A steel shed is to be constructed, with dimensions as shown below. The shed is to include a rectangular cut-out at the front for the entrance.

a

Determine the surface area of the shed. Round your answer to one decimal place.

b

Construction of the shed requires an additional 0.1\text{ m}^2 of sheet metal for each 1\text{ m}^2 of surface area, due to overlaps and wastage.

How much sheet metal is required to construct this shed? Round your answer up to the nearest square metre.

c

If the steel sheets cost \$18 per square metre, calculate the total cost of the steel required to build this shed.

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Solve problems involving surface area and volume for a range of prisms, cylinders and composite solids.

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