Question

A wire of length 28 cm is to be cut into two pieces. One of the two pieces is to be made into a square and the other into a circle. What should be the length of two pieces so that the combined area of them is minimum?

Answer 1

Let the length of one piece be x cm, then the length of other piece will be (28 - x) cm.

Let from the first piece we make a circle of radius r and from the second piece we make a square of side y.

Then,

Let A be the combined area of the circle and square then,

A = πr² + y²

⇒ A = \(\pi(\frac{x}{2\pi})^2+\)\((\frac{28-x}{4})^2\) ...(ii)

Differentiating (ii) and with respect to 'x', we get

For maximum and minimum A' = 0

Since, 

A' = + ve for \(x= \frac{28\pi}{4+\pi}\)

∴ A is min for \(x= \frac{28\pi}{4+\pi}\)

Thus, the required length of two pieces are

\(x= \frac{28\pi}{4+\pi}\) cm and 28 - \(x\)

= 28 - \(\frac{28\pi}{4+\pi}\)

= \(\frac{192}{4+\pi}\)cm.