Optimal Values for Composite Functions - Word Problem

[afroman]

A woman lives on an island 2km from the mainland. Her fitness club is 4km along the short from the point closest to the island. To get to the fitness club, she paddles her kayak at 2km/h. Once she reaches the shore, she jogs at 4km/h. Determine where she should land to reach her fitness club in the shortest possible time.

I'm having trouble with this questions, can anyone help me please?
Let t represent the time in hours.
d=sqrt of (2-2t)sqrd+(4-4t)sqrd
Is that the right starting equation?
Thanks

 

[galactus]

Draw a diagram. That would help.

Distance equals rate times time. So you could use t=d/r.

t=time from island to point on shore+ time from point on shore to fitness club.

\(\displaystyle \L\\r_{p}=rate \;\ paddling, \;\ r_{j}=rate \;\ jogging\)

\(\displaystyle \L\\t=\frac{\sqrt{x^{2}+4}}{r_{p}}+\frac{4-x}{r_{j}}\)

You are given the rates of the paddling and jogging.

Differentiate, set to 0 and solve for x.

 

[afroman]

I don't think I'm getting the right answer as it doesn't make sense. What did you get and could you show me the steps? Your help is appreciated. Thanks.

 

[galactus]

What did you get?. The answer I arrived at seems plausible.

 

[afroman]

I got x=.5 and y=5.6. What did you get and how? I know I'm doing some little mistake. Thanks

 

[stapel]

I got x=.5 and y=5.6....

How?

Please reply showing all of your work and reasoning. Thank you.

Eliz.

 

[afroman]

t = sqrt x^2+4 + 4-x
derivative of t = 1/2(x^2+4)^-1/2(2x)+(-1)
= 2x-1/2 sqrt x^2+4
solve for x: 2x-1=0
x=1/2
sub x into t: t= sqrt (1/2)^2+4 + 4-(1/2)
t= 5.6

This is telling me that the quickest way is 1/2hour and you would have to land 5.6km down the coast...which of course does not make sense in this context. Thanks for everyones help.

 

[galactus]

What are you differentiating?.

Use the quotient rule.

\(\displaystyle \L\\\frac{d}{dx}\left[\frac{\sqrt{x^{2}+4}}{2}\right]=\frac{x}{2\sqrt{x^{2}+4}}\)

\(\displaystyle \L\\\frac{d}{dx}\left[\frac{4-x}{4}\right]=\frac{-1}{4}\)


Now, finish?.