A wire of length \((\pi + 4)\) cm is cut into two pieces. One piece is bent into a circle and the other into a square. For the sum of the areas of the circle and the square to be least, the length of the piece bent into the circle must be
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\pi\) cm
Let the piece for the circle be x cm. The circle has radius \(\frac{x}{2\pi}\), and the square has side \(\frac{\pi + 4 - x}{4}\). The total area is
\( \displaystyle A(x) = \pi\left(\frac{x}{2\pi}\right)^{2} + \left(\frac{\pi + 4 - x}{4}\right)^{2} \)
\( \displaystyle = \frac{x^{2}}{4\pi} + \frac{(\pi + 4 - x)^{2}}{16} \)
Setting the derivative equal to zero:
\( \displaystyle A'(x) = \frac{x}{2\pi} - \frac{\pi + 4 - x}{8} = 0 \)
\( 4x = \pi(\pi + 4 - x) \)
\( (4 + \pi)x = \pi(\pi + 4) \), so \(x = \pi\)
Since \( \displaystyle A''(x) = \frac{1}{2\pi} + \frac{1}{8} \gt 0 \), this is a minimum. The circle piece is π cm (radius \(\frac{1}{2}\)) and the square piece is 4 cm (side 1), giving the least total area \(\frac{\pi}{4} + 1 \approx 1.785\) cm2.
Why the other options are incorrect
Option (1) 4 cm — Incorrect. 4 cm is the length of the piece bent into the square, not the circle.
Option (3) \(\displaystyle\frac{\pi + 4}{2}\) cm — Incorrect. Cutting the wire into equal halves gives a total area of about 1.811 cm2, more than 1.785.
Option (4) \(\displaystyle\frac{\pi}{2}\) cm — Incorrect. This gives a total area of about 2.136 cm2, more than 1.785.