IB Maths AI HL/Question Bank/3.1 Geometry and trigonometry - SL content

IB Maths AI HL 3.1 Geometry and trigonometry - SL content Question Bank

HL58 questions10 previewsSyllabus linked

[Maximum number: 15]

A farmer owns a field in the shape of a triangle ABC such that $\mathrm{AB}=650 \mathrm{~m}, \mathrm{AC}=1005 \mathrm{~m}$ and $\mathrm{BC}=1225 \mathrm{~m}$ .

(a)

Find the size of $A \hat{C} B$ .

The local town is planning to build a highway that will intersect the borders of the field at points D and E , where $\mathrm{DC}=210 \mathrm{~m}$ and $\mathrm{CED}=100^{\circ}$ , as shown in the diagram below.

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(b)

Find DE .

The town wishes to build a carpark here. They ask the farmer to exchange the part of the field represented by triangle DCE. In return the farmer will get a triangle of equal area ADF, where F lies on the same line as D and E, as shown in the diagram above.

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(c)

Find the area of triangle DCE.

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(d)

Estimate DF. You may assume the highway has a width of zero.

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[Maximum number: 6]

A child's game is played by making paths in the snow. First, two circular paths are made using the same centre, O . The radius of the smaller circle is 2.8 m , and the radius of the larger circle is 4 m . Additional paths are then made from O to the outer edge of the larger circle, dividing each circle into 5 equal sectors, as shown in the following diagram.
For your calculations, ignore the widths of the paths.

Each region between paths will be coloured with environmentally friendly dye. The shaded region below will be coloured orange.

Maureen has enough orange dye to cover an area of $6 \mathrm{~m}^{2}$ .

(a)

Show that Maureen has enough orange dye to cover the shaded region.

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(b)

During the game, the players start at the entrance and must travel only along the paths made in the snow. Maureen travels from the entrance along the path shown in the following diagram.

Calculate the distance Maureen travels along this path, starting from the entrance and returning to the entrance.

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[Maximum number: 7]

Two schools are represented by points A(2,20) and B(14,24) on the graph below. A road, represented by the line R with equation -x+y=4, passes near the schools. An architect is asked to determine the location of a new bus stop on the road such that it is the same distance from the two schools.

(a)

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

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(b)

Determine the coordinates of the point on R where the bus stop should be located.

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[Maximum number: 13]

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 $\mathrm{OA}=25.9 \mathrm{~m}$ and $\mathrm{OAB}=125^{\circ}$ .

(a)

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 $=45^{\circ}$ .

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(b)

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 $\mathrm{O} \hat{\mathrm{K}} \mathrm{X}=51.1^{\circ}$ .

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(c)

Find whether Odette or Khemil is closer to the football.

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

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(d)

Calculate the distance that Khemil runs.

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[Maximum number: 3]

In this question you will use a historic method of calculating the cost of a barrel of wine to determine which shape of barrel gives the best value for money.
In Austria in the 17th century, one method for measuring the volume of a barrel of wine, and hence determining its cost, was by inserting a straight stick into a hole in the side, as shown in the following diagram, and measuring the length SD . The longer the length, the greater the cost to the customer.

Let SD be d metres and the cost be C gulden (the local currency at the time). When the length of SD was 0.5 metres, the cost was 0.80 gulden.

(a)

This method of determining the cost was noticed by a mathematician, Kepler, who decided to try to calculate the dimensions of a barrel which would give the maximum volume of wine for a given length SD.

Initially he modelled the barrel as a cylinder, with S at the midpoint of one side. He took the length of the cylinder as 2 h metres and its radius as r metres, as shown in the following diagram of the cross-section.

Find an expression for $r^{2}$ in terms of d and h.

Let the volume of this barrel be $V \mathrm{~m}^{3}$ .

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[Maximum number: 9]

Three points N, P, and V are shown on the following diagram. NP is 20 metres, PV is 25 metres and VPN is $55.5^{\circ}$ .

(a)

Find NV.

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(b)

Find PN̂V.

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(c)

Hence or otherwise, find the shortest distance between P and [NV].

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[Maximum number: 12]

This question considers the optimal route between two points, separated by several regions where different speeds are possible.
Huw lives in a house, H , and he attends a school, S , where H and S are marked on the following diagram. The school is situated 1.2 km south and 4 km east of Huw's house. There is a boundary [MN], going from west to east, 0.4 km south of his house. The land north of [MN] is a field over which Huw runs at 15 kilometres per hour $\left(\mathrm{km} \mathrm{h}^{-1}\right)$ . The land south of [MN] is rough ground over which Huw walks at $5 \mathrm{~km} \mathrm{~h}^{-1}$ . The two regions are shown in the following diagram.

diagram not to scale

(a)

Huw travels in a straight line from H to S . Calculate the time that Huw takes to complete this journey. Give your answer correct to the nearest minute.

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(b)

(i)

For the optimal route, verify that the equation in part (c)(ii) satisfies the following result:

\frac{\cos \mathrm{H} \hat{\mathrm{P} M}}{\cos \mathrm{~S} \hat{\mathrm{P}} \mathrm{~N}}=\frac{\text { speed over field }}{\text { speed over rough ground }}

(c)

The owner of the rough ground converts the southern quarter into a field over which Huw can run at $15 \mathrm{~km} \mathrm{~h}^{-1}$ . The following diagram shows the optimal route, HJKS, in this new situation. You are given that [HJ] is parallel to [KS].

diagram not to scale

Using a similar result to that given in part (c)(iii), at the point J, determine MJ.

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[Maximum number: 3]

Kailash manufactures drink containers in the shape of a cuboid. The container has a square top and a square base of length, $l \mathrm{~cm}$ . Its height, $d \mathrm{~cm}$ , is three times the length of the base.

(a)

Calculate the total external surface area of the container.

To reduce environmental impact, Kailash is trying to minimize the amount of material needed for the production of the $375 \mathrm{~cm}^{3}$ container.

He is willing to change the shape to a cylinder with radius $r \mathrm{~cm}$ , and height $h \mathrm{~cm}$ , as shown below.

The cylindrical container of drink must also hold $375 \mathrm{~cm}^{3}$ .

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[Maximum number: 18]

This question considers how the assessment of the Air Quality Index (AQI) for a school depends on the method chosen by the person doing the assessing.
Air quality for a district is measured at three monitoring stations. The positions of these stations on a coordinate system with units in kilometres are A(0,5), B(8,9) and C(8,1).
A Voronoi diagram is constructed with the three stations as sites.

(a)

(i)

Find the equation of the perpendicular bisector of [BC].

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(ii)

Given that the equation of the perpendicular bisector of $[\mathrm{AB}]$ is y=15-2 x, find the coordinates of vertex V.

A school, S, is situated in the district at the point with coordinates (5,6).

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(b)

State which station is closest to the school.

The principal of the school is concerned about the air quality in the area. Air quality is measured by the Air Quality Index (AQI). In this district, values less than 50 are taken to indicate good air quality.

The principal contacts the local environmental agency requesting an AQI value for her school. They tell her the mean AQI reading from the closest station to the school.

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(c)

Write down the type of interpolation being used by the environmental agency.

The principal obtains the mean AQI value from each of the three stations.

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(d)

(i)

Find the mid-point of [SB].

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(ii)

Show that the equation of the perpendicular bisector of $[\mathrm{SB}]$ is y=14-x.

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(iii)

Hence find $a_{B}$ (the area of region Q on the diagram).

The areas of regions P and R are $a_{A}=13.7 \mathrm{~km}^{2}$ and $a_{C}=6.9 \mathrm{~km}^{2}$ respectively.

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(e)

Use the natural neighbour algorithm to show that an estimate for the expected AQI value at the school, W, is 94.4.

The principal is still concerned that this method is underestimating the AQI value at the school, as the school is situated close to a busy traffic intersection. She decides to take her own readings ( x ) over a period of 60 days. Her results are summarized as

\bar{x}=97.8, s_{n-1}=17.2, n=60 .

The principal assumes that the daily AQI values at the school can be modelled by a normal distribution and that each value is independent of any other value.

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[Maximum number: 18]

Madhu is designing a jogging track for the campus of her school. The following diagram shows an incomplete portion of the track.
Madhu wants to design the track such that the inner edge is a smooth curve from point A to point B , and the other edge is a smooth curve from point C to point D . The distance between points A and B is 50 metres.

To create a smooth curve, Madhu first walks to M , the midpoint of $[\mathrm{AB}]$ .

(a)

Madhu then walks 20 metres in a direction perpendicular to [A B] to get from point M to point F. Point F is the centre of a circle whose arc will form the smooth curve between points A and B on the track, as shown in the following diagram.

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(i)

Find the length of $[\mathrm{BF}]$ .

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(ii)

Find BFA .

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(b)

Hence, find the length of arc AB .

The outer edge of the track, from C to D, is also a circular arc with centre F, such that the track is 2 metres wide.

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(c)

Calculate the area of the curved portion of the track, ABDC .

The base of the track will be made of concrete that is 12 cm deep.

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(d)

Calculate the volume of concrete needed to create the curved portion of the track.

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