Need help solving this AP Calc problem...? - water taxi between maii and kona
An island is at point A, 6 km from the coast of B, the closest point to the right on a beach. A woman on the island, to the point C, 9 km along the beach as you can take a boat for $ 2.50 per mile travel and rent of water at a point P between B and C and can take a taxi to a price of $ 2.00 per kilometer and the trip to take right path from P to C. Find the best route from point A to point C.
Saturday, February 20, 2010
Water Taxi Between Maii And Kona Need Help Solving This AP Calc Problem...?
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A
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BP
Let x be the acute angle between the segments AB and AP.
Let us assume that the length of B P.
Let B be the length of P to C
(We know that a + b = 9)
Let c be the length of the A to P
The cost function is 2.5 * c + 2 * B.
Since the line AB has a length of 6, 6 / c = cos (x).
Therefore, c = 6sec (x)
And c ^ 2 = 6 ^ 2 + a ^ 2, by the theorem of Pythagoras.
Thus 36sec ^ 2 (x) = 36 + a ^ 2, and so
6sqrt a = (s ^ 2 (x) -1) = 6sqrt (tan ^ 2 (x)) = 6tan (x).
Therefore, B = 9 to 6 * tan (x).
Now we can rewrite the cost function as
f (x) = 2.5 * 6 * s (x) + 2 * (9 up to 6 * tan (x)) = 15 sec (x) - 12tan (x) + 18.
Take a derivative:
f '(x) = 15 sec (x) tan (x) - 12 sec ^ 2 (x)
Setting it to 0 and the cancellation of a sec (x) gives
15tan (x) = 12 sec (x)
15sin (x) / cos (x) = 12/cos (x)
15sin (x) = 12
sin (x) = 12/15 = 4 / 5
After plugging in x = sin ^ -1 (4 / 5) in the objective function, I'm 11, which means that 11 is the minimum cost.
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