Kindly provide the answer in detail (with full working) Sania\'s boat is at poin

Question

Kindly provide the answer in detail (with full working)

Sania's boat is at point A on the sea. The closest point on land, point B, is 2 km. away. Point C on land is 6 km. from point B, such that triangle (ABC) is right-angled at point B. Sanya wishes to reach point C, by rowing to some point P on the line BC, and jog the remaining distance to C. If she rows 2 km. per hour and jogs 5 km. per hour, at what distance from point B should she choose her landing point P, in order to minimize her time to reach point C? (a) 21/squareroot 4 (b) 4/squareroot 21 (c) 4/squareroot 12 (d) 21/squareroot 21

Explanation / Answer

Sania rows at 2km per hour, so she will take 1/2(4+x2)1/2 hour to reach the point P on the shore. Further,

Sania jogs at 5 km per hour so that she will take (6-x)/5 hour to reach C from P. Hence, the total time

taken by Sania to reach the point C from A is t = 1/2(4+x2)1/2 + (6-x)/5. If t is to be minimum, then dt/dx =

0 and d2t/dx2 has to be positive. Here, dt/dx = (1/2)[ (1/2)(4+x2)-1/2 * 2x] -1/5 . Thus if dt/dx = 0, then

(1/2)[ (1/2)(4+x2)-1/2* 2x] -1/5 = 0 or, x/2(4+x2)-1/2= 1/5 or, x(4+x2)-1/2= 2/5 or, x2/(4+x2) = 4/25 or, 25x2=

4(4+x2) or, 25x2= 4x2+16 or, 21x2= 16 so that x = 21/4. Further, d2t/dx2= x/2[ (-1/2) (4+x2)-3/2*2x] +

½[ (4+x2)-1/2 ] = ½[ (4+x2)-1/2 ] - x/4 [ (4+x2)-3/2 = ½( 4+x2)-3/2 [ (4+x2)- x] which is positive when x = 21/4.

Hence, the total time from the point A to the point C will be minimum if the distance of the point P from B is 21/4 km. The option (a) is the correct answer.

  

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