Problem Solving

A jogging park has two identical circular tracks touching each other, and a
rectangular track enclosing the two circles. The edges of the rectangles are
tangential to the circles. Two friends, A and B, start jogging simultaneously from the
point where one of the circular tracks touches the smaller side of the rectangular
track. A jog along the rectangular track, while B jogs along the two circular tracks in
a figure of eight. Approximately, how much faster than A does B have to run, so that
they take the same time to return to their starting point?

Ans: 4.72%

Since they both take same time to reach their starting point , the ratio of their speeds will be ratio of the distance travelled.



Distance travelled by A = 12R (perimeter of rectangle.)
Distance travelled by B = 4 π R

Speeds of A : Speed of B = 12R:4Ï€R = > 3:Ï€

The question asks Approximately, how much faster than A does B have to run, so that
they take the same time to return to their starting point

(Ï€-3)/3 * 100.

I followed a similar concept, but used numbers to make my math easier, instead of straight algebra.

Lets use easy numbers:

Length of rectangle = 8 (Since, this will give each circle a diameter of 4 and radius of 2. Nice whole numbers)

Since the diameter is 4, the width of the rectangle has to be 4

Permiter of rectangle = 8 + 8 + 4 + 4 = 24

Circumfrence of a circle = 2(3.14)(2) = 12.56

Circumfrence of both circles = 25.12

24 is 96% of 25.

That means the person running on the circles has to walk little faster than 4% of the pace of the person running on the rectangle.