# At noon a small plane left an airport and flew due east at 200 km/h. Two hours later another plane left the same airport and flew due south at 400 km/h. At what time will the planes be exactly 800 km apart from one another?

Jul 25, 2018

color(blue)("Both planes will be 800 km apart after " 3 hrs 12 min " or at "15 hrs 12 min " or " 3 : 12 p.m.

#### Explanation:

Let x be the time in hours taken for the planes be 800 km apart.

$\text{First plane would have travelled " (200 * x) km " in x hours.}$

$\text{Second plane would have travelled " (400 * (x-2)) km " after x hours as it had started 2 hours behind.}$

Since first plane has left towards east and the second towards south, they moved in perpendicular directions and we can conveniently apply Pythagoras Theorem.

${\left(200 \cdot x\right)}^{2} + {\left(400 \cdot \left(x - 2\right)\right)}^{2} = {800}^{2}$

${200}^{2} \cdot \left({x}^{2} + {\left(2 \left(x - 2\right)\right)}^{2}\right) = {800}^{2}$

${x}^{2} + 4 {x}^{2} - 16 x + 16 = {\left(\frac{800}{200}\right)}^{2} = 16$

$5 {x}^{2} - 16 x + \cancel{16} - \cancel{16} = 0$

$5 {\cancel{{x}^{2}}}^{\textcolor{red}{x}} = 16 \cancel{x}$

$5 x = 16 \text{ or } x = 3.2 h o u r s = 3 h r s 12 \min$#