A wire of length 28 cm is cut into two pieces. One piece is bent into a square and the other into a circle. The combined area of the square and the circle is minimum when the length of the piece used for the circle is:
A wire of length 28 cm is cut into two pieces. One piece is bent into a square and the other into a circle. The combined area of the square and the circle is minimum when the length of the piece used for the circle is:
- A. 28π/(π + 4) cm
- B. 28/(π + 4) cm
- C. 28π/(π + 2) cm
- D. 14π/(π + 4) cm
Answer: A) 28π/(π + 4) cm
Explanation: Let length for circle = x, radius r = x/(2π). Length for square = 28 − x, side = (28 − x)/4. Total area A = π(x/(2π))² + ((28 − x)/4)² = x²/(4π) + (28 − x)²/16. dA/dx = x/(2π) − (28 − x)/8 = 0 → multiply by 8π: 4x − π(28 − x) = 0 → 4x − 28π + πx = 0 → x(4 + π) = 28π → x = 28π/(4 + π).
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