es work Sets ?5: Problem 7 roblem 7 ettings Previous Problem Problem List Next P

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es work Sets ?5: Problem 7 roblem 7 ettings Previous Problem Problem List Next Problem (1 point) Problems Brandon is on one side of a river that is 70 m wide and wants to reach a point 300 m downstream on the opposite side as quickly as possible by swimming diagonally across the river and then running the rest of the way. Find the minimum amount of time if Brandon can swim at 2 m/s and run at 4 m/s. (Use decimal notation. Give your answer to two decimal places.) em 3 m 5 The minimum amount of time is seconds help (fractions) Preview My Answers Submit Answers You have attempted this problem 0 times. You have unlimited attempts remaining. 30

Explanation / Answer

If x is the distance Brandon runs, then the time spent running is
.. tr = x/4 ... seconds.
The time spent swimming is
.. ts = (1/2)?(70^2 + (300-x)^2)
so his total travel time is
.. t = tr + ts = x/4 + 1/2 ?(x^2 - 600x + 94900)

This will be minimized when its derivative with respect to x is zero.
.. 0 = dt/dx = 1/4 + (2x-600)/(4?(x^2 - 600x + 94900)
.. -?(x^2 - 600x + 94900) = 2x - 600
.. x^2 - 600x + 94900 = 4x^2 - 2400x + 360000
.. 3x^2 - 1800x + 265100 = 0
.. x = (1800 - ?58800)/6 ? 259.58
When the time expression is evaluated for this value of x, the result is
.. t = 105.31 ... seconds

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