An airoplane which flying along the sides of a square at speed 300,400,500,600 km/hr find the average speed .justify your answer

To find the average speed of the airplane flying along the sides of a square, we can use the following approach: 1. **Understand the Path and Speed**: The airplane is flying along the perimeter of a square. Let's denote the side length of the square as \( s \). 2. **Perimeter Calculation**: The perimeter \( P \) of a square with side length \( s \) is given by: \[ P = 4s \] 3. **Distance Travelled at Each Speed**: The airplane travels each side of the square which is \( s \) long. Assuming each side is travelled at a different speed, we can set the speeds for the four sides of the square as follows: - Side 1 at speed \( v_1 = 300 \) km/hr - Side 2 at speed \( v_2 = 400 \) km/hr - Side 3 at speed \( v_3 = 500 \) km/hr - Side 4 at speed \( v_4 = 600 \) km/hr 4. **Time Taken to Travel Each Side**: The time taken to travel each side is calculated as: \[ t_1 = \frac{s}{v_1}, \quad t_2 = \frac{s}{v_2}, \quad t_3 = \frac{s}{v_3}, \quad t_4 = \frac{s}{v_4} \] This gives: \[ t_1 = \frac{s}{300}, \quad t_2 = \frac{s}{400}, \quad t_3 = \frac{s}{500}, \quad t_4 = \frac{s}{600} \] 5. **Total Time**: The total time \( T \) taken to complete the circuit is: \[ T = t_1 + t_2 + t_3 + t_4 = \frac{s}{300} + \frac{s}{400} + \frac{s}{500} + \frac{s}{600} \] 6. **Finding a Common Denominator**: The least common multiple of \( 300, 400, 500, \) and \( 600 \) is \( 1200 \). Thus, we can rewrite the fractions: \[ T = s \left( \frac{4}{1200} + \frac{3}{1200} + \frac{2.4}{1200} + \frac{2}{1200} \right) \] \[ T = s \left( \frac{4 + 3 + 2.4 + 2}{1200} \right) = s \left( \frac{11.4}{1200} \right) \] 7. **Distance**: The total distance travelled is the perimeter: \[ D = 4s \] 8. **Average Speed**: The average speed \( V_{\text{avg}} \) is defined as total distance divided by total time: \[ V_{\text{avg}} = \frac{D}{T} = \frac{4s}{\frac{11.4s}{1200}} \] \[ V_{\text{avg}} = \frac{4s \cdot 1200}{11.4s} \] \[ V_{\text{avg}} = \frac{4800}{11.4} \approx 421.05 \text{ km/hr} \] Thus, the average speed of the airplane flying along the sides of the square is approximately **421.05 km/hr**. ### Justification The average speed takes into account the different speeds at which the airplane travels each side of the square. The overall average is influenced more by the sections where the airplane was slower (i.e., at 300 km/hr and 400 km/hr) than the sections where it was faster (500 km/hr and 600 km/hr). This means that simply averaging the speeds would not accurately reflect the true average speed due to varying time durations at each speed.