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# Calculus problem help?

Bob is at point B 25 miles from A. Alice is in a boat in the sea at point C 1.75 miles from the beach. Alice rows at 2.5 miles per hour and walks at 4 miles per hour , where along the beach should she land so that she may get to Bob in the least amount of time ?

### 2 Answers

- M3Lv 71 decade agoFavorite Answer
.. C

.. |\ angle ACP = z.

.. |-\

w|--\

...|----\

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

..A. x P ...B

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

let speed ratio land:water = k (4/2.5)

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)

plugging in,

x = 1.75/√(1.6^2 - 1)

ans: 1.401 km to 3 dp [ calculation error rectified ]

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note that the optimal value of angle z is solely dependent on k !

- ted sLv 71 decade ago
DRAW the figure...let x = distance along the beach where boat beaches , tb = boat time and tw = walk time....time = tb + tw = [ √(1.75² + x²) / 2.5 ] + [ (25 - x ) / 4 ]...find x which minimizes time...you do the computations

{ (7/4) / √15 }....lot of walking....sure that Bob is 25 miles away ??