# Two boats travel at right angles to each other after leaving the same dock at the same time. 1 hour later they are 5 miles apart. If one travels 1 miles faster than the other, what is the rate of each?

Feb 16, 2017

Faster boat: 4 miles/hr; Slower boat: 3 miles/hr

#### Explanation:

Let the slower boat travel at $x$ miles/hr

$\therefore$ the faster boat travels at $\left(x + 1\right)$ miles/hr

After 1 hour the slower boat has travelled $x$ miles
and the faster boat has travelled $x + 1$ miles.

We are told that:
(i) the boats travel at right angles to each other and
(ii) after 1 hour the boats are 5 miles apart

Hence we can use Pythagoras on the right angle triangle formed by the path of both boats and the distance between them as follows:

${x}^{2} + {\left(x + 1\right)}^{2} = {5}^{2}$

${x}^{2} + {x}^{2} + 2 x + 1 = 25$

$2 {x}^{2} + 2 x - 24 = 0$

${x}^{2} + x - 12 = 0$

$\left(x + 4\right) \left(x - 3\right) = 0$

Since: $x > 0 \to x = 3$

$\therefore$ The faster boat travels at $\left(3 + 1\right) = 4$ miles/hr; The slower boat travels at 3 miles/hr.