Math Help - Theoretical rate problem

1. Theoretical rate problem

I have finished this problem and got some answers, but I don't think there right. If someone can confirm my answers or correct them that would be a big help.

A woman lives on an island that is 1 km from the mainland. She paddles her canoe at 3 km/h and jogs at 5 km/h. She would like to go to the drug store that is 3 km along the shore from the point on the shore closest to the island. Where should she land to reach the drug store in minimum time?

There are three parts I did. First I found the time it would take for her to canoe straight across and jog the rest of the way and got 0.56 hours. Next I calculated her canoeing straight to the drug store and got 1.05 hours. The last part would be somewhere inbetween and I got 0.612 hours. If all of that is correct then her landing point should be 52.1 m from the drug store.

Please if anyone can confirm if my answers are correct or show me your solution with explination if mine is wrong, it would help me out significantly.

2. Originally Posted by dcfi6052
...

A woman lives on an island that is 1 km from the mainland. She paddles her canoe at 3 km/h and jogs at 5 km/h. She would like to go to the drug store that is 3 km along the shore from the point on the shore closest to the island. Where should she land to reach the drug store in minimum time?

...
Hello,

I've attached a rough sketch of the situation.

$speed=\frac{distance}{time}$. Thus

$time=\frac{distance}{speed}$

First calculate the distances:

$s=\sqrt{1+x^2}$ is the distance which is travelled by canoe.

$l = 3 \text{ km} - x$ is the distance to jog.

Now use the formula to calculate the time needed:

$t(x)=\frac{\sqrt{1+x^2}}{3}+\frac{3-x}{5}$

This is a function of t in x. Calculate the first derivative:

$t'(x)=\frac{1}{6} \cdot (1+x^2)^{-\frac{1}{2}} \cdot 2x - \frac{1}{5}$. To get the minimum t'(x) = 0:

$\frac{x}{\sqrt{1+x^2}}=\frac{3}{5}$. Square the equation and multiply by (1+x²). You'll get $\frac{16}{25} x^2=\frac{9}{25}$

The only possible result is x = 0.75

Plug in this value in t(x) and you get the minimum time: 0.86666 ... = 13/15 hours = 52 minutes.

EB