The diagram shows a field ABCDE.
(a) Calculate the perimeter of the field ABCDE.
(b) Calculate angle ABD.
(c) (i) Calculate angle CBD.
(ii) The point C is due north of the point B.
Find the bearing of D from B.
(d) Calculate the area of the field ABCDE.
Give your answer in hectares.
[1 hectare = 10 000m2]
▶️ Answer/Explanation
(a) Ans: 530 m
Perimeter = Sum of all sides: AB (150) + BC (80) + CD (90) + DE (120) + EA (90).
(b) Ans: 52.9°
Using cosine rule in triangle ABD: cos(ABD) = (150² + 120² – 130²)/(2×150×120).
(c)(i) Ans: 28.1°
Using sine rule in triangle BCD: sin(CBD) = (90×sin(120°))/130.
(c)(ii) Ans: 331.9°
Bearing = 360° – (angle ABD + angle CBD) = 360° – (52.9° + 28.1°).
(d) Ans: 1.5 hectares
Total area = Area of triangle ABD (using ½absinC) + Area of triangle BCD (using ½absinC).
Brad travelled from his home in New York to Chamonix.
- He left his home at 1630 and travelled by taxi to the airport in New York.
This journey took 55 minutes and had an average speed of 18km/h. - He then travelled by plane to Geneva, departing from New York at 2215.
The flight path can be taken as an arc of a circle of radius 6400km with a sector angle of 55.5°.
The local time in Geneva is 6 hours ahead of the local time in New York.
Brad arrived in Geneva at 1125 the next day. - To complete his journey, Brad travelled by bus from Geneva to Chamonix.
This journey started at 1300 and took 1 hour 36 minutes.
The average speed was 65km/h.
The local time in Chamonix is the same as the local time in Geneva.
Find the overall average speed of Brad’s journey from his home in New York to Chamonix.
Show all your working and give your answer in km/h.
▶️ Answer/Explanation
Ans: 392.5 km/h
Step 1: Calculate taxi distance: \( \text{Distance} = \text{Speed} \times \text{Time} = 18 \times \frac{55}{60} = 16.5 \, \text{km} \).
Step 2: Compute flight distance (arc length): \( L = r \times \theta = 6400 \times \left( \frac{55.5 \times \pi}{180} \right) \approx 6200 \, \text{km} \).
Step 3: Find bus distance: \( \text{Distance} = 65 \times \frac{96}{60} = 104 \, \text{km} \).
Step 4: Total distance = \( 16.5 + 6200 + 104 = 6320.5 \, \text{km} \).
Step 5: Total time = 16 hours 6 minutes = 16.1 hours. Average speed = \( \frac{6320.5}{16.1} \approx 392.5 \, \text{km/h} \).