# Math Help - Optimization

1. ## Optimization

A snail, who can travel at 3 km/h hour is 10 km from the nearest point P on a straight highway. He wishes to travel to a point Q, 50 km along the highway from P, and she can arrange to have a turtle pick her up anywhere along the highway. The turtle travels at 15 km/h. What point on the highway should the snail head for as to arrive at Q in the shortest possible time?

Wouldn't the snail want to be picked up at point P to get there the quickest?

2. LOL. loved your problem, it was so cute.

anyway yes you are right. it should be picked up at point P.

3. Originally Posted by steezin
A snail, who can travel at 3 km/h hour is 10 km from the nearest point P on a straight highway. He wishes to travel to a point Q, 50 km along the highway from P, and she can arrange to have a turtle pick her up anywhere along the highway. The turtle travels at 15 km/h. What point on the highway should the snail head for as to arrive at Q in the shortest possible time?

Wouldn't the snail want to be picked up at point P to get there the quickest?
not at point P.

make a sketch.

let x = distance from point P where the snail hitches the ride.

distance snail travels overland ... $\sqrt{x^2+10^2}
$

distance snail travels on the highway with turtle ... $50-x$

$t = \frac{\sqrt{x^2+10^2}}{3} + \frac{50-x}{15}
$

minimize $t$