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Calculate the distance from M to the top of the tower at T

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IB Questionbank Mathematical Studies 3rd edition 1

1. An old tower (BT) leans at 10 away from the vertical (represented by line TG).

The base of the tower is at B so that MBˆT = 100.

Leonardo stands at L on flat ground 120 m away from B in the direction of the lean.

He measures the angle between the ground and the top of the tower T to be BLˆT = 26.5.

M MB = 200 B G BL = 120

26.5°

not to scale T

90° L 100°

(a) (i) Find the value of angle BTˆL.

(ii) Use triangle BTL to calculate the sloping distance BT from the base, B to the top, T of the tower.

(5) (b) Calculate the vertical height TG of the top of the tower.

(2) (c) Leonardo now walks to point M, a distance 200 m from B on the opposite side of the

tower. Calculate the distance from M to the top of the tower at T.

(3) (Total 10 marks)

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IB Questionbank Mathematical Studies 3rd edition 2

2. The diagram below shows a field ABCD with a fence BD crossing it. AB = 15 m, AD = 20 m and angle BAˆD = 110°. BC = 22 m and angle BDˆC = 30°.

110°

30°

15 20

22

A

B

C

D

diagram not to scale

(a) Calculate the length of BD.

(3)

(b) Calculate the size of angle BCˆD.

(3)

One student gave the answer to (a) “correct to 1 significant figure” and used this answer to calculate the size of angle BCˆD.

(c) Write down the length of BD correct to 1 significant figure.

(1)

(d) Find the size of angle BCˆD that the student calculated, giving your answer correct to 1 decimal place.

(2)

(e) Hence find the percentage error in his answer for angle BCˆD.

(3) (Total 12 marks)

(3)

IB Questionbank Mathematical Studies 3rd edition 3

3. (a) A farmer wants to construct a new fence across a field. The plan is shown below. The new fence is indicated by a dotted line.

75°

40°

410 m

Diagram not to scale

Calculate the length of the fence.

(5)

(b) The fence creates two sections of land. Find the area of the smaller section of land ABC, given the additional information shown below.

24245 m

Diagram not to scale

A B

C

(3) (Total 8 marks)

(4)

IB Questionbank Mathematical Studies 3rd edition 4

4. A recreation park has two trains. Train 1 takes visitors from the entrance (E) to the swimming pool (S), to the mini golf (M) and back to the entrance. Train 2 takes visitors from the entrance (E) to the play area (P), to the racing track (R) and back to the entrance. This is shown in the diagram.

115°

50°

TRAIN 2

TRAIN 1

E S

M P

R

750 m

400 m

500 m ES = 500 m

SM = 400 m ER = 750 m ESM = 115°

ERP = 50°

EPR = 90°

[not to scale]

(a) Calculate the total distance Train 2 travels in one journey from E to P to R to E.

(5)

(b) (i) Show that EM = 761 m correct to 3 s.f..

(ii) If the trains travel at 2 ms–1 find the time taken for Train 1 to complete a journey from E to S to M to E. Give your answer to the nearest second.

(6) (Total 11 marks)

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