For fencing of flower bed with 100 cm long wire in the form of circular sector, the maximum area of the flower bed is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 625 cm2
Let the sector have radius r cm and arc length l cm. The wire runs along the two radii and the arc:
\(2r + l = 100\)
\(l = 100 - 2r\)
The area of a sector is half the radius times the arc length:
\( \displaystyle A = \frac{1}{2}rl = \frac{1}{2}r(100 - 2r) \)
\(A = 50r - r^2\)
For the maximum, set the derivative to zero:
\( \displaystyle \frac{dA}{dr} = 50 - 2r = 0 \)
\(r = 25\)
\( \displaystyle \frac{d^2A}{dr^2} = -2 \lt 0 \)
so r = 25 gives a maximum. Then
\(A = 50(25) - 25^2 = 1250 - 625 = 625\)
The maximum area is 625 cm2.