IB Maths AI SL/Question Bank/3 Geometry and trigonometry

IB Maths AI SL 3 Geometry and trigonometry Question Bank

SL83 questions10 previewsSyllabus linked

[Maximum number: 17]

The diagram shows points in a park viewed from above, at a specific moment in time.
The distance between two trees, at points A and B , is 6.36 m .
Odette is playing football in the park and is standing at point O , such that $\mathrm{A} \hat{\mathrm{O}} \mathrm{B}=10^{\circ}$ , $\mathrm{OA}=25.9 \mathrm{~m}$ and OAB is obtuse.

(a)

Calculate the size of ABO .

[ 3 ]

(b)

Calculate the area of triangle AOB .

Odette's friend, Khemil, is standing at point K such that he is 12 m from A and KAB $=45^{\circ}$ .

[ 4 ]

(c)

Calculate Khemil's distance from B.

XY is a semicircular path in the park with centre A , such that $\mathrm{KA} \mathrm{Y}=45^{\circ}$ . Khemil is standing on the path and Odette's football is at point X . This is shown in the diagram below.

The length $\mathrm{KX}=22.2 \mathrm{~m}, \mathrm{KOPX}=53.8^{\circ}$ and $\mathrm{O} \hat{\mathrm{K}} \mathrm{X}=51.1^{\circ}$ .

[ 3 ]

(d)

Find whether Odette or Khemil is closer to the football.

Khemil runs along the semicircular path to pick up the football.

[ 4 ]

(e)

Calculate the distance that Khemil runs.

[ 3 ]

[Maximum number: 5]

The front view of a doghouse is made up of a square with an isosceles triangle on top. The doghouse is 1.35 m high and 0.9 m wide, and sits on a square base.

The top of the rectangular surfaces of the roof of the doghouse are to be painted.
Find the area to be painted.

[Maximum number: 4]

Kacheena stands at point K , the top of a 218 m vertical cliff. The base of the cliff is located at point B. A ship is located at point $S, 1200 \mathrm{~m}$ from Kacheena.
This information is shown in the following diagram.

(a)

Find the angle of elevation from the ship to Kacheena.

[ 2 ]

(b)

Find the horizontal distance from the base of the cliff to the ship.

[ 2 ]

[Maximum number: 4]

One of the steepest train tracks in the world is in Tennessee, USA.
This track is 1.52 km long, and the angle of elevation from the bottom of the track to the top is $36.1^{\circ}$ .

(a)

Label the diagram with the given values for the track length and the angle of elevation.

[ 2 ]

(b)

Find the vertical change in height from the bottom of the track to the top.

[ 2 ]

[Maximum number: 5]

A metal structure on a flat surface is in the form of a right-pyramid with rectangular base ABCD and vertex H(-0.5,1.5,5). Point A has coordinates (-2,0,0) and point C has coordinates (1,3,0). This is shown in the following diagram.
All units are in centimetres.

The centre of the base, G, is the midpoint of A C.

(a)

Find the coordinates of G.

[ 2 ]

(b)

Write down the vertical height HG.

[ 1 ]

(c)

Find the distance between C and H .

[ 2 ]

[Maximum number: 5]

The straight metal arm of a windscreen wiper on a car rotates in a circular motion from a pivot point, O , through an angle of $140^{\circ}$ . The windscreen is cleared by a rubber blade of length 46 cm that is attached to the metal arm between points A and B . The total length of the metal arm, OB , is 56 cm .
The part of the windscreen cleared by the rubber blade is shown unshaded in the following diagram.

(a)

Calculate the length of the arc made by B , the end of the rubber blade.

[ 2 ]

(b)

Determine the area of the windscreen that is cleared by the rubber blade.

[ 3 ]

[Maximum number: 4]

One of the steepest straight roads in the world is in Dunedin, New Zealand.
This road is 161 m long, and the angle of elevation from the bottom of the road to the top is $16.3^{\circ}$ .

(a)

Label the diagram with the given values for the road length and the angle of elevation.

[ 2 ]

(b)

Find the vertical change in height from the bottom of the road to the top.

[ 2 ]

[Maximum number: 8]

The following grid shows a restaurant's floorplan. There are four food stations centred at points P, Q, R and S . The Voronoi diagram for these four points is shown. Point Q is not shown.

One unit represents 1 metre.
Point P is located at (2,2).
The equation of the perpendicular bisector of $[\mathrm{PQ}]$ is y=4.

(a)

Write down the coordinates of Q.

Points R and S are located at ( 11,0 ), and ( 12,6 ), respectively.

[ 1 ]

(b)

Find

[ 6 ]

(i)

the coordinates of the midpoint of [RS]

(ii)

the equation of the perpendicular bisector of $[\mathrm{RS}]$ . Give your answer in the form y=m x+c.

Customers in the restaurant take their food from the nearest food station.

The following table shows the average waiting time, in minutes, for each food station.

[ 6 ]

(c)

Using nearest-neighbour interpolation, find the average waiting time for a customer at point (6,8).

The restaurant owner wishes to determine whether customers spend more money during the weekend. She decides to use a two-sample t-test at a 5 % level of significance.

For this test, the null hypothesis is:

\mathrm{H}_{0}: \mu_{1}=\mu_{2}

where $\mu_{1}$ is the mean amount of money spent by all customers on weekdays, and $\mu_{2}$ is the mean amount of money spent by all customers on weekends.

[ 1 ]

[Maximum number: 14]

The following grid shows a restaurant's floorplan. There are four food stations centred at points A, B, C and D . The Voronoi diagram for these four points is shown. Point A is not shown.

One unit represents 1 metre.
Point B is located at (2,6).
The equation of the perpendicular bisector of $[\mathrm{AB}]$ is y=4.

(a)

Write down the coordinates of A.

Points C and D are located at (9,0), and (10,6), respectively.

[ 1 ]

(b)

Find

[ 12 ]

(i)

the coordinates of the midpoint of [CD]

[ 6 ]

(ii)

the equation of the perpendicular bisector of $[\mathrm{CD}]$ . Give your answer in the form y=m x+c.

Customers in the restaurant take their food from the nearest food station.

The following table shows the average waiting time, in minutes, for each food station.

[ 6 ]

(c)

Using nearest-neighbour interpolation, find the average waiting time for a customer at point (4,6).

The restaurant owner wishes to determine whether customers spend more money during the weekend. She decides to use a two-sample t-test at a 5 % level of significance.

For this test, the null hypothesis is:

\mathrm{H}_{0}: \mu_{1}=\mu_{2}

where $\mu_{1}$ is the mean amount of money spent by all customers on weekdays, and $\mu_{2}$ is the mean amount of money spent by all customers on weekends.

[ 1 ]

[Maximum number: 4]

A vertical pole stands on horizontal ground. The bottom of the pole is taken as the origin, O, of a coordinate system in which the top, F , of the pole has coordinates (0,0,5.8). All units are in metres.

The pole is held in place by ropes attached at F.
One of the ropes is attached to the ground at a point A with coordinates (3.2,4.5,0). The rope forms a straight line from A to F.

(a)

Find the length of the rope connecting A to F .

[ 2 ]

(b)

Find FA A , the angle the rope makes with the ground.

[ 2 ]