Question

Points $A$ and $B$ are $120$ km apart. A motor-cyclist starts from $A$ to $B$ along a straight road $AB$ with the speed of $30$ km/h. At the same time, a cyclist starts from $B$ along a road, which is perpendicular to $AB$ with the speed of $10$ km/h. When will the distance between them be the least?

Answer 1

The distance between two motor cyclist would be length of hyper use which is square root of

${(120-3x)}^2+{\left(10x\right)}^2={1000x}^2-60\times 120x+{120}^2$

$\Rightarrow {1000x}^2-60\times 120x+{120}^2$

Now quadratic fraction $f\left(x\right)={ax}^2+bx+c$ reaches the minimum or maximum when a is negative.

When $x=\frac{-b}{2a}=\frac{60\times 120}{2\times 1000}=3.6$.