Question
A running track of 440 ft is to be laid out enclosing a football field, the shape of which is a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, then the lengths of its side are
BITSAT
Mathematics
Application of Derivatives
2022
BITSAT 2022
BITSAT Mathematics Question (2022) — Solution
Options
- A. 70 ft and 110 ft
- B. 80 ft and 120 ft
- C. 35 ft and 110 ft
- D. 35 ft and 120 ft
Answer
C. 35 ft and 110 ft
Step-by-step solution
Perimeter =440 ft 2 x+ r+ r=440 2 x+2 r=440 aligned A & = Area of the rectangular portion =x 2 r \\ A & =x (440-2 x) = 1 (440 x-2 x^2 ) aligned Let d A d x = 1 (440-4 x)=0 aligned & x=110 for which d^2 A d x^2 < 0 \\ & A is maximum when x=110 \\ & 2 r= 440-2 x = 440-220 22 / 7 =70 \\ & r=35 ft and x=110 ft aligned
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