The total length of the wire is 2 units. This wire is cut into two parts:

One part of length $4x$ used to form a square of side $x$.

The other part of length $2\pi r$ used to form a circle of radius $r$.

The perimeter constraint is:

$ 4x + 2\pi r = 2. $

Express one variable in terms of the other:

From the constraint, we can express $x$ in terms of $r$:

$ 4x = 2 - 2\pi r \implies x = \frac{2 - 2\pi r}{4} = \frac{1 - \pi r}{2}. $

Write the area function:

The total area $A$ is the sum of the area of the square and the area of the circle:

$ A = \text{(Area of square)} + \text{(Area of circle)} = x^2 + \pi r^2. $

Substitute $x = \frac{1 - \pi r}{2}$ into the area expression:

$ A = \left(\frac{1 - \pi r}{2}\right)^2 + \pi r^2. $

Expand the square:

$ \left(\frac{1 - \pi r}{2}\right)^2 = \frac{(1 - \pi r)^2}{4} = \frac{1 - 2\pi r + \pi^2 r^2}{4}. $

Thus:

$ A = \frac{1 - 2\pi r + \pi^2 r^2}{4} + \pi r^2. $

Combine like terms:

$ A = \frac{1}{4} - \frac{2\pi r}{4} + \frac{\pi^2 r^2}{4} + \pi r^2. $

Factor where possible:

$ A = \frac{1}{4} - \frac{\pi r}{2} + \left(\frac{\pi^2 r^2}{4} + \pi r^2\right). $

Combine the $r^2$ terms:

$ \frac{\pi^2 r^2}{4} + \pi r^2 = \frac{\pi^2 r^2 + 4\pi r^2}{4} = \frac{\pi(\pi+4)r^2}{4}. $

Hence:

$ A = \frac{1}{4} - \frac{\pi r}{2} + \frac{\pi(\pi+4)r^2}{4}. $

Find the critical point for minimization:

Differentiate $A$ with respect to $r$:

$ A'(r) = -\frac{\pi}{2} + \frac{\pi(\pi+4)}{2}r. $

Set $A'(r) = 0$ to find the critical point:

$ -\frac{\pi}{2} + \frac{\pi(\pi+4)}{2}r = 0. $

Divide through by $\frac{\pi}{2}$ (nonzero):

$ -1 + (\pi+4)r = 0 \implies (\pi+4)r = 1 \implies r = \frac{1}{\pi+4}. $

Find the corresponding $x$:

Substitute $r = \frac{1}{\pi+4}$ into $x = \frac{1-\pi r}{2}$:

$ x = \frac{1 - \pi\left(\frac{1}{\pi+4}\right)}{2} = \frac{1 - \frac{\pi}{\pi+4}}{2} = \frac{\frac{\pi+4-\pi}{\pi+4}}{2} = \frac{4}{(\pi+4) \cdot 2} = \frac{2}{\pi+4}. $

We now have:

$ x = \frac{2}{\pi+4}, \quad r = \frac{1}{\pi+4}. $

Check the given options:

Option A: $2x = (\pi+4)r$

Substitute values:

$ 2x = 2 \cdot \frac{2}{\pi+4} = \frac{4}{\pi+4}, \quad (\pi+4)r = (\pi+4)\cdot \frac{1}{\pi+4} = 1. $

Not equal.

Option B: $(4-\pi)x = \pi r$

Substitute:

$ (4-\pi)x = (4-\pi)\frac{2}{\pi+4}, \quad \pi r = \pi\frac{1}{\pi+4}. $

This does not simplify to a known identity matching our values.

Option C: $x = 2r$

Substitute:

$ 2r = 2\cdot\frac{1}{\pi+4} = \frac{2}{\pi+4}, \quad x = \frac{2}{\pi+4}. $

This matches perfectly. $x = 2r$.

Option D: $2x = r$

Substitute:

$ 2x = 2\cdot\frac{2}{\pi+4} = \frac{4}{\pi+4}, \quad r = \frac{1}{\pi+4}. $

Not equal.

The correct relation is $ x = 2r $.

Final Answer:

$ \boxed{x = 2r} \text{ (Option C)} $

Application Of Derivatives

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