Power Line Optimization QUestion? (calculus)?? HELP URGENT!?

Question

A man lives on an island 1 km from the mainland. His favorite pub is 3 km along the shore from the point on the shore closest to the island. The man can paddle his canoe at 3 km/h and can jog at 5 km/h. Determine where he should land so as to reach the pub in the shortest possible time.

I really don't understand it. PLZ HELP! it would be greatly appreciated!!

Answer 1

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.. C island

.. |\ angle ACP = z.

.. |-\

w|--\

...|----\

.. |-----\------

..A. x P ...B pub

...<---- l --->

let speed ratio land:water = k (5/3)

dist. over water = w•sec z,

dist. over land = l - w•tan z

time taken ∞ k•sec z +(l - tan z)

dT/dz = k•secz•tan z - sec^2 z

= (ksin z - 1) / cos^2 z

for minima, sin z = 1/k

which yields x = w /√(k^2 - 1)

x = 1/√((5/3)^2 - 1)

ans: 0.75 km

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Answer 2

the better proper nook is at (x, 5-x^2) the better left nook is at (-x, 5-x^2) so the width is 2x the height is 5-x^2 So the area is wh = 10x - 2x^3 take the deriv and set to 0 10 = 6x^2 x^2 = 10/6 = 5/2 clean up for x about a million.fifty 8 height = 5-x^2: calculate and multiply for section