A RUNNING TRACK OF 440 FT IS LAID OUT ENCLOSING A FOOTBALL FIELD, THE SHAPE OF WHICH IS RECTANGLE WITH - Maths - Application of Derivatives

Question

A RUNNING TRACK OF 440 FT IS LAID OUT ENCLOSING A FOOTBALL FIELD,
THE SHAPE OF WHICH IS RECTANGLE WITH SEMICIRCLE AT EACH END IF THE
AREA OF THE RECTANGULAR PORTION IS MAXIMUM FIND THE LENGTH OF THE
SIDES??

Garima Singhal · Accepted Answer

Let x be the length of rectangular portion and y be the diameter of semicircular ends
Now, perimeter of track = 440 ft⇒x + x + 2π×y2 = 440⇒2x + πy = 440⇒y = 440-2xπ    
1Now, area of rectangular portion = xyA = xyA = x 440-2xπA = 1π440x-2x2dAdx = 1π440-4xNow, for maxima or minimadAdx = 0⇒1π440-4x = 0⇒x = 110Now d2Adx2 = 1π0-4 = -4π < 0So, Area is maximum at x = 110Now, y = 440 - 220π = 220×722 = 70So, x = 110 ft and y = 70 ft

A RUNNING TRACK OF 440 FT IS LAID OUT ENCLOSING A FOOTBALL FIELD, THE SHAPE OF WHICH IS RECTANGLE WITH SEMICIRCLE AT EACH END. IF THE AREA OF THE RECTANGULAR PORTION IS MAXIMUM FIND THE LENGTH OF THE SIDES??