Maggie and Aggie synchronize their watches, rehearse their plan one last time, and start driving their cars at exactly midnight. Maggie heads due east, and Aggie heads due north, traveling 15 mph faster than Maggie. At exactly 1:20 a.m. the two women are exactly 100 miles apart, as observed from an inconspicuous police glider by Sulphronia the super agent, who is watching through infra-red binoculars. At what speed are the two sisters driving?

## Solution

Enter Pythagoras: $A^2 + M^2 = 100^2$. The driving time is 1 hour 20 minutes, or 4/3 hours.

Maggie has gone $\frac{4}{3}x$ miles; Aggie has gone $\frac{4}{3}(x + 15)$ miles = $\frac{4}{3}x + 20$ miles.

So $(\frac{4}{3}x)^2 + (\frac{4}{3}x + 20)^2 =100^2$

= $\frac{16x^2}{9} + \frac{16x^2}{9} + \frac{160x}{3} + 400 = 10,000$

$\rightarrow \frac{32x^2}{9} + \frac{160x}{3} = 9600 \rightarrow 32x^2 + 480x = 86,400$

$\rightarrow x^2 + 15x - 2700 = 0 \rightarrow (x + 60)(x - 45) = 0$

$\rightarrow x = 45$ and that's Maggie. And Aggie is going at 60 mph.

Lucky factoring! Or use the quadratic formula.

One wonders just what their plan was.