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# Calculus question!!! see below...?

A dune buggy is on a straight desert road, 40 km north of Dustin City. The

vehicle can travel at 45 km/h off the road and 75 km/h on the road. The

driver wants to get to Gulch City, 50 km east of Dustin by another straight

road, in the shortest possible time. Determine the route he should take.

### 2 Answers

- M3Lv 710 years agoFavorite Answer
.. C

.. |\ angle ACP = z.

.. |-\

w|--\

...|----\

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

..A. x P ...B gulch city

...<---- y --->

let dist. to dustin city = w (40 km)

speed ratio regular road: desert road = k (5/3)

dist. over desert = w•sec z,

dist. over regular road = y - w•tan z

time taken ∞ k•sec z +(y - 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 = 40/√((5/3)^2 - 1)

ans: x = 30 km

------

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

- stollsteimerLv 45 years ago
Its (eleven+five)/(eleven+15) =sixteen/26 = sixty one.538% roughly. (Note: I'm assuming that it entails those who're freshman "and" feminine. So, if thats now not the case, my reply is flawed then. Sorry.)