Calculating Aeroplane Average Speed Over Equilateral Triangle

The average speed is defined as the total distance covered divided by the total time taken.

Problem Setup

• The aeroplane travels along the three sides of an equilateral triangle.
• Let the length of each side of the equilateral triangle be d kilometres.
• The speeds for each side are: $v_1 = 300$ km/h, $v_2 = 200$ km/h, and $v_3 = 240$ km/h.

Step-by-Step Solution

1. Calculate Total Distance:

   The total distance is the perimeter of the equilateral triangle.

   Total Distance = $d + d + d = 3d$ km.

2. Calculate Time Taken for Each Side:

   Time = Distance / Speed

   • Time for side 1 ($t_1$) = $\frac{d}{v_1} = \frac{d}{300}$ hours.
   • Time for side 2 ($t_2$) = $\frac{d}{v_2} = \frac{d}{200}$ hours.
   • Time for side 3 ($t_3$) = $\frac{d}{v_3} = \frac{d}{240}$ hours.
3. Calculate Total Time:

   Total Time = $t_1 + t_2 + t_3 = \frac{d}{300} + \frac{d}{200} + \frac{d}{240}$ hours.

   To add these fractions, find a common denominator. The least common multiple (LCM) of 300, 200, and 240 is 1200.

   Total Time = $\frac{4d}{1200} + \frac{6d}{1200} + \frac{5d}{1200} = \frac{(4+6+5)d}{1200} = \frac{15d}{1200}$ hours.

   Simplify the fraction: Total Time = $\frac{d}{80}$ hours.

4. Calculate Average Speed:

   Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}}$

   Average Speed = $\frac{3d}{\frac{d}{80}}$

   Average Speed = $3d \times \frac{80}{d}$

   The distance '$d$' cancels out.

   Average Speed = $3 \times 80 = 240$ km/h.

Conclusion

The average speed of the aeroplane while flying around the triangle is 240 km/h.