Question

I’m stumped, because I only see the most efficient path being walking to point B then swimming to C

1. In an Ironman competition a competitor arrives at point A with the goal of reaching point C in the least possible time. She can run along the side of the lake toward point B at a steady speed of 9.00 km/h, and she can swim in the lake at a steady speed of 3.00 km/h. Calculate the 4.00 miles LAKE distance from point A to the point where she should she start swimming in order to achieve her goal. 5.00 miles

Answer 1

Let x be the distance he ran from A

the time taken in running this distance

t1 = x/9

The width of the lake = 4 miles = 6.4 km

The lenght along the lake = 5 mil = 8km

where swimming starts is (8-x) km

Using pythagoras theorem will give distance of swimming

$\sqrt{(8-x)^2 + 6.4^2}$

t2 = $\sqrt{(8-x)^2 + 6.4^2}$ / 3

t = t1 + t2

= x/9 + $\sqrt{(8-x)^2 + 6.4^2}$ /3

to make this time minimum, we use principle of minima

dt/dx = 0

1/9 - 2(8-x)/6$\sqrt{(8-x)^2 + 6.4^2}$ = 0

solving for x in calci

x = 7.947 km