question_answer

An aeroplane leaves an airport and flies due North at a speed of \[300\,km\] per hour. At the same time, another aeroplane leaves the same airport and flies due West at a speed of \[500\,km\] per hour. How far apart will be two planes after \[1\frac{1}{2}\] hours?

A) \[200\sqrt{29}\,km\]

B) \[100\sqrt{3}\,km\]

C) \[150\sqrt{34}\,km\]

D) \[250\sqrt{34}\,km\]

Correct Answer: C

Solution :

[c]  Distance \[=\text{ Speed}\times \text{Time}\]

Distance travelled by the aeroplane due north in \[1\frac{1}{2}\] hour

\[=300\times \frac{3}{2}=450\,km\]

Distance travelled by the aeroplane due west in \[1\frac{1}{2}\] hour

\[=500\times \frac{3}{2}=750\,km\]

In \[\Delta AOB,\] by Pythagoras theorem

\[A{{B}^{2}}=A{{O}^{2}}+O{{B}^{2}}\]

\[\Rightarrow \,\,A{{B}^{2}}={{(450)}^{2}}+{{(750)}^{2}}\]

\[\Rightarrow \,\,A{{B}^{2}}=202500+562500=765000\]

\[\therefore \,\,\,\,\,AB=\sqrt{765000}\]

\[=\sqrt{9\times 25\times 34\times 100}=150\sqrt{34}km\]