A calculus problem by Vishal Sharma

Calculus Level 2

Consider a man standing at point 'P' at a distance 'a' units from the wall. This man has to reach at a point 'Q' (which is on the same side of the wall). Distance between 'P' & 'Q' , measured along the wall is 'b' units , while the distance of 'Q' from the wall is 'c' units. The man has to touch the wall before arriving at point 'Q'.

Locate the point on the wall that the man should touch before arriving at 'Q' , so that the distance covered by him is the least ??

Let the distance of the point be 'x' units . FIND 'x'

DETAILS AND ASSUMPTIONS: a = 1 units b = 25 units c = 4 units

assume the man to travel with a constant speed..

note: TRY TO DO IT BY SOME OTHER WAY THAN CALCULUS...


The answer is 5.

Vishal Sharma by Vishal Sharma , 26, Germany

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4 solutions

Kuntal Das
Mar 30, 2014

Without Calculus. Just in one step..

Take mirror image of Q about x axis. Now we know displacement is equal to shortest distance btw 2 points. So connect source and the image of Q. Now apply simmilarity of triangles.

(x/a)=[b/(a+c)]

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Arpit Chahande
Mar 27, 2014

without using calculus: the distance will be shortest if the hypotenuse of both the triangles have the same ratio as the sides.... that means both the sides must be of the ratio a:c and hence we can equate the ratio of one side to hypotenuses for both the triangles the formulation becomes (squared) x^2/(a^2+x^2)={(b-x)^2}/{c^2+(b-x)^2}

the equation reduces to a simple quadratic equation .. giving answer 5

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Awnon Bhowmik
Mar 27, 2014

Easy does it. Plain old similar triangles. Back to high school guys :)

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Vishal Sharma
Mar 26, 2014

A short and nice method to solve it could be: assume man first lights a torch towards the wall and light reflects from the wall to the point where man has to reach.

since we very well know that light chooses the shortest possible path. the man can then follow the path of light....

Apply laws of reflection and find 'x'

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pls share your methods of solving this ques...

Vishal Sharma - 7 years, 2 months ago

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draw the mirror image of the smaller triangle considering wall as mirror.for shortest distance the path should be a straight line.so the slope shall be same.so 1/x=4/20 hence x=5

PRASHANT KUMAR - 7 years, 2 months ago

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The method I proposed is just like what you said... Cause when you apply law of reflection you get the mirror image of P on extrapolating..

Vishal Sharma - 7 years, 2 months ago

why does light choose the shortest possible path?

Tanay Prateek - 6 years, 3 months ago

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