A man is camping on an island 1 km from the nearest point on the shoreline. Assume that the shoreline is a straight line and perpendicular to the line, which represents the shortest between the island and the shoreline. He wants to go to the ranger’s office, which is 4 km down the shoreline. He can paddle his canoe at 5 km/h and he can run at 10 km/h, toward what point on shore should he aim his canoe in order to reach the office in the least time?

Question

sshayer · Answer

|dw:1468032106347:dw|

iroman12 · Answer

Can you show it to me how would you deal with this problem using calculus?

mww · Answer

Yes this one you need calculus to associate the running and paddling distances

mww · Answer

|dw:1468036935995:dw|

Sketch above. AC can be determined using Pythagoras to be
$$AC = \sqrt{1+x^2}$$

mww · Answer

Now the time for each event (paddle, run) is given by the formula
t = distance/speed.
So we can write two equations to represent the time for paddling and running
$$t_{paddle} = \frac{ distance_{paddle} }{ speed_{paddle} } = \frac{ \sqrt{1+x^2} }{ 5 }$$
$$t_{run} = \frac{ distance_{run} }{ speed_{run} } = \frac{ 4 -x  }{ 10 }$$

So the total time for the entire journey is 
$$t = \frac{ \sqrt{1+x^2} }{  5} + \frac{ 4-x }{ 10 }$$
The smallest time occurs when you differentiate and find x for which 
$$\frac{ dt }{ dx } = 0 $$

sshayer · Answer

|dw:1468079236938:dw|