Geometry and Trigonometry
Name:________________________________________________________
1a. [4 marks]
A right pyramid has apex
and rectangular base
, with
. The vertical height of the pyramid is
Calculate
.
1b. [2 marks]
Calculate the volume of the pyramid.
1
.
,
and
2a. [2 marks]
Fabián stands on top of a building, T, which is on a horizontal street.
He observes a car, C, on the street, at an angle of depression of 30°. The base of the building is at B. The
height of the building is 80 metres.
The following diagram indicates the positions of T, B and C.
Show, in the appropriate place on the diagram, the values of
(i) the height of the building;
(ii) the angle of depression.
2b. [2 marks]
Find the distance, BC, from the base of the building to the car.
2c. [2 marks]
Fabián estimates that the distance from the base of the building to the car is 150 metres. Calculate the
percentage error of Fabián’s estimate.
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3a. [1 mark]
In the following diagram, ABCD is the square base of a right pyramid with vertex V. The centre of the
base is O. The diagonal of the base, AC, is 8 cm long. The sloping edges are 10 cm long.
Write down the length of
.
3b. [2 marks]
Find the size of the angle that the sloping edge
makes with the base of the pyramid.
3c. [3 marks]
Hence, or otherwise, find the area of the triangle
.
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4a. [6 marks]
The following diagram shows a perfume bottle made up of a cylinder and a cone.
The radius of both the cylinder and the base of the cone is 3 cm.
The height of the cylinder is 4.5 cm.
The slant height of the cone is 4 cm.
(i) Show that the vertical height of the cone is
cm correct to three significant figures.
(ii) Calculate the volume of the perfume bottle.
4b. [2 marks]
The bottle contains
of perfume. The bottle is not full and all of the perfume is in the cylinder
part.
Find the height of the perfume in the bottle.
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4c. [4 marks]
Temi makes some crafts with perfume bottles, like the one above, once they are empty. Temi wants to
know the surface area of one perfume bottle.
Find the total surface area of the perfume bottle.
4d. [4 marks]
Temi covers the perfume bottles with a paint that costs 3 South African rand (ZAR) per millilitre. One
millilitre of this paint covers an area of
.
Calculate the cost, in ZAR, of painting the perfume bottle. Give your answer correct to two decimal
places.
4e. [2 marks]
Temi sells her perfume bottles in a craft fair for 325 ZAR each. Dominique from France buys one and
wants to know how much she has spent, in euros (EUR). The exchange rate is 1 EUR = 13.03 ZAR.
Find the price, in EUR, that Dominique paid for the perfume bottle. Give your answer correct to two
decimal places.
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5a. [2 marks]
In triangle
,
,
and
.
diagram not to scale
Find the length of
.
5b. [2 marks]
is the point on
Find the length of
such that
.
.
5c. [2 marks]
is the point on
such that
Find the area of triangle
.
.
6
6a. [3 marks]
A manufacturer has a contract to make
triangular prism,
solid blocks of wood. Each block is in the shape of a right
, as shown in the diagram.
and angle
Calculate the length of
.
.
6b. [3 marks]
Calculate the area of triangle
.
6c. [2 marks]
Assuming that no wood is wasted, show that the volume of wood required to make all
, correct to three significant figures.
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blocks is
6d. [2 marks]
Write
in the form
where
and
.
6e. [3 marks]
Show that the total surface area of one block is
, correct to three significant figures.
6f. [3 marks]
The blocks are to be painted. One litre of paint will cover
Calculate the number of litres required to paint all
blocks.
8
.
7a. [1 mark]
A greenhouse ABCDPQ is constructed on a rectangular concrete base ABCD and is made of glass. Its
shape is a right prism, with cross section, ABQ, an isosceles triangle. The length of BC is 50 m, the length
of AB is 10 m and the size of angle QBA is 35°.
Write down the size of angle AQB.
7b. [3 marks]
Calculate the length of AQ.
7c. [2 marks]
Calculate the length of AC.
7d. [2 marks]
Show that the length of CQ is 50.37 m, correct to 4 significant figures.
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7e. [3 marks]
Find the size of the angle AQC.
7f. [5 marks]
Calculate the total area of the glass needed to construct
(i) the two rectangular faces of the greenhouse;
(ii) the two triangular faces of the greenhouse.
7g. [3 marks]
The cost of one square metre of glass used to construct the greenhouse is 4.80 USD.
Calculate the cost of glass to make the greenhouse. Give your answer correct to the nearest 100 USD.
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8a. [3 marks]
A tent is in the shape of a triangular right prism as shown in the diagram below.
The tent has a rectangular base PQRS .
PTS and QVR are isosceles triangles such that PT = TS and QV = VR .
PS is 3.2 m , SR is 4.7 m and the angle TSP is 35°.
Show that the length of side ST is 1.95 m, correct to 3 significant figures.
8b. [3 marks]
Calculate the area of the triangle PTS.
8c. [1 mark]
Write down the area of the rectangle STVR.
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8d. [3 marks]
Calculate the total surface area of the tent, including the base.
8e. [2 marks]
Calculate the volume of the tent.
8f. [4 marks]
A pole is placed from V to M, the midpoint of PS.
Find in metres,
(i) the height of the tent, TM;
(ii) the length of the pole, VM.
8g. [2 marks]
Calculate the angle between VM and the base of the tent.
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