Racing Track Design

00:00 15 marks

Ansel is designing a racing track for a local bicycle club. The following diagram shows an incomplete portion of the track.

Ansel 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 90 metres.

[VISUALIZATION – Diagram showing two parallel track sections connected by a bend. Points A and C are on the inner and outer edges of one straight section. Points B and D are on the inner and outer edges of the other straight section. M is the midpoint of AB. The distance AB is marked as 90 metres.]

diagram not to scale

To create a smooth curve, Ansel first walks to M, the midpoint of [AB].

(a) Write down the length of [BM]. [1]

Ansel then walks 32 metres in a direction perpendicular to [AB] to get from point M to point E. Point E 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.

[VISUALIZATION – Diagram showing the bend from the previous visualization. Point E is shown such that ME is perpendicular to AB. Lines EB and EA are drawn, forming triangle ABE with E as the vertex opposite base AB. ME = 32m.]

diagram not to scale

(b) (i) Find the length of [BE].
(ii) Find BÊM. [4]

The outer edge of the track, from C to D, is also a circular arc with centre E, such that the track is 4 metres wide.

(d) Calculate the area of the curved portion of the track, ABDC. [4]

The base of the track will be made of concrete that is 15 cm deep.

(e) Calculate the volume of concrete needed to create the curved portion of the track. [3]

Hint Hint

Hint

For part (a) use the midpoint concept (BM = 90/2). For part (b) apply the Pythagorean theorem to find BE and then use tan(BÊM) = BM/ME to find the angle. For part (c) double the angle from part (b) and use the arc length formula (arc length = radius × angle). For part (d), calculate the area of two sectors (with radii BE and BE+4) and find the difference. For part (e), convert 15 cm to 0.15 m and multiply by the area.

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Answer Answer

Complete Answer

Part (a): BM = 45 m.
Part (b)(i): BE = √(45² + 32²) = √3049 ≈ 55.2 m.
Part (b)(ii): BÊM = arctan(45/32) ≈ 0.953 radians (≈ 54.6°).
Part (c): Arc AB = BE × (2 × BÊM) ≈ 55.2 × 1.906 ≈ 105 m.
Part (d): Outer radius = BE + 4 ≈ 59.2 m; area = (1/2) × ((59.2)² – (55.2)²) × 1.906 ≈ 435 m².
Part (e): Volume = 435 m² × 0.15 m ≈ 65.3 m³.

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