Geometry and trigonometryIB Maths: Applications and Interpretation SL: Topic test

1. 1
   A glass paperweight is a solid hemisphere of radius 6 cm.

   (a)

   Find the volume of the paperweight.

   [1 mark]

   • A$905$ cm³
   • B$339$ cm³
   • C$452$ cm³
   • D$226$ cm³

   (b)

   Find the total surface area of the paperweight, including its flat circular face.

   [1 mark]

   • A$226$ cm²
   • B$339$ cm²
   • C$452$ cm²
   • D$679$ cm²

   (c)

   The glass has density $2.5$ g cm⁻³. Find the mass of the paperweight, in grams, correct to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A rotating lawn sprinkler stands at the centre $O$ of a circular lawn. The water reaches 9 m from $O$, and the sprinkler waters a sector of the lawn by turning through an angle of $200^\circ$.

   (a)

   Find the area of lawn that is watered.

   [1 mark]

   • A$254$ m²
   • B$31.4$ m²
   • C$565$ m²
   • D$141$ m²

   (b)

   Find the length of the outer edge of the watered region, the arc of the sector.

   [1 mark]

   • A$31.4$ m
   • B$56.5$ m
   • C$15.7$ m
   • D$141$ m

   (c)

   A thin pipe is laid along the whole boundary of the watered region: the arc and the two straight edges. Find the length of the pipe.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A Voronoi diagram is drawn for two wind turbines at the sites $A(1,6)$ and $B(9,2)$. Coordinates are in km.

   (a)

   Find the equation of the edge that separates the cell of $A$ from the cell of $B$. Give your answer in the form $y=mx+c$.

   [3 marks]

   (b)

   A weather sensor is placed at the point on this edge that also lies on the line $x+y=12$. Find the coordinates of the sensor, and find its distance from turbine $A$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A building has a roof in the shape of a right pyramid $VABCD$. The base $ABCD$ is a horizontal rectangle with $AB=12$ m and $BC=8$ m. The vertex $V$ is 5 m vertically above the centre of the base.

   (a)

   (i) Find the length of the diagonal $AC$.
   (ii) Find the length of the sloping edge $VA$.
   (iii) Find the angle between $VA$ and the horizontal base.
   Give lengths in metres and angles in degrees, each correct to 3 significant figures.

   [6 marks]

   (b)

   The point $M$ is the midpoint of $[AB]$ and $N$ is the midpoint of $[BC]$.
   (i) Find the length $VM$.
   (ii) Hence find the area of the sloping face $VAB$.
   (iii) Find the area of the sloping face $VBC$, and hence the total area of the four sloping faces of the roof.
   Give lengths in metres and areas in m², each correct to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A Voronoi diagram is drawn for three fire stations at the sites $A(0,0)$, $B(10,0)$ and $C(2,8)$. Coordinates are in km.

   (a)

   Which equation gives the edge between the cell of $A$ and the cell of $B$?

   [1 mark]

   • A$y=5$
   • B$x=5$
   • C$x=10$
   • D$y=0$

   (b)

   A fire starts at the point $(6,1)$. Find the distance from the fire to the nearest fire station.

   [1 mark]

   • A$6.08$ km
   • B$8.06$ km
   • C$17$ km
   • D$4.12$ km

   (c)

   The edge between the cells of $A$ and $C$ lies on the line $x+4y=17$. Find the coordinates of the vertex where the edges between $A$, $B$ and $C$ meet.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A triangular sail has sides of length 6.5 m, 4.8 m and 7.2 m.

   (a)

   Find the size of the largest angle of the sail.

   [1 mark]

   • A$77.6^\circ$
   • B$40.6^\circ$
   • C$102^\circ$
   • D$61.8^\circ$

   (b)

   Find the area of the sail.

   [1 mark]

   • A$30.5$ m²
   • B$15.6$ m²
   • C$15.2$ m²
   • D$13.7$ m²

   (c)

   Find the size of the smallest angle of the sail, correct to 3 significant figures.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A coastguard station $C$ observes a ship $S$ at a distance of 8 km on a bearing of $040^\circ$. A second station $D$ is 7 km due east of $C$.

   (a)

   Find the distance $DS$, in kilometres, correct to 3 significant figures.

   [3 marks]

   (b)

   Find the bearing of the ship from station $D$. Give your answer as a three-figure bearing to the nearest degree.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A paper cone is made by cutting a sector of radius 20 cm and angle $216^\circ$ from a circle of paper and joining its two straight edges together with no overlap. The arc of the sector becomes the circumference of the base of the cone.

   (a)

   (i) Find the length of the arc of the sector.
   (ii) Find the radius of the base of the cone.
   (iii) Find the vertical height of the cone.
   Give lengths in centimetres, correct to 3 significant figures.

   [6 marks]

   (b)

   (i) Find the volume of the cone.
   (ii) Find the curved surface area of the cone.
   (iii) Find the angle between the sloping side of the cone and its base.
   Give the volume in cm³, the area in cm² and the angle in degrees, each correct to 3 significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions