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

Options

A. `L_1= 112/(pi+4)`, `L_1= (28pi)/(pi+4)`
B. `L_1= 142/(pi+4)`, `L_1= (28pi)/(pi+4)`
C. `L_1= 112/(pi+4)`, `L_1= pi/(pi+4)`
D. None of these

Explanation

To solve the problem of cutting a wire of length 28 m into two pieces to minimize the combined area of a square and a circle, we can follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = length of the wire piece used to form the square
- \( R \) = length of the wire piece used to form the circle

### Step 2: Set Up the Perimeter Equation
The perimeter of the square is \( 4x \) and the circumference of the circle is \( 2\pi R \). Since the total length of the wire is 28 m, we can write:
\[ 4x + 2\pi R = 28 \]

### Step 3: Express R in Terms of x
Rearranging the perimeter equation gives:
\[ 2\pi R = 28 - 4x \]
Dividing by \( 2\pi \):
\[ R = \frac{28 - 4x}{2\pi} = \frac{14 - 2x}{\pi} \]

### Step 4: Set Up the Area Function
The area \( A \) of the square is \( x^2 \) and the area of the circle is \( \pi R^2 \). Thus, the total area \( A \) can be expressed as:
\[ A = x^2 + \pi R^2 \]
Substituting \( R \):
\[ A = x^2 + \pi \left( \frac{14 - 2x}{\pi} \right)^2 \]
Simplifying this:
\[ A = x^2 + \pi \cdot \frac{(14 - 2x)^2}{\pi^2} = x^2 + \frac{(14 - 2x)^2}{\pi} \]

### Step 5: Expand the Area Function
Expanding \( (14 - 2x)^2 \):
\[ (14 - 2x)^2 = 196 - 56x + 4x^2 \]
Thus, the area function becomes:
\[ A = x^2 + \frac{196 - 56x + 4x^2}{\pi} = \left(1 + \frac{4}{\pi}\right)x^2 - \frac{56}{\pi}x + \frac{196}{\pi} \]

### Step 6: Differentiate the Area Function
To find the minimum area, we differentiate \( A \) with respect to \( x \):
\[ \frac{dA}{dx} = 2\left(1 + \frac{4}{\pi}\right)x - \frac{56}{\pi} \]

### Step 7: Set the Derivative to Zero
Setting the derivative equal to zero to find critical points:
\[ 2\left(1 + \frac{4}{\pi}\right)x - \frac{56}{\pi} = 0 \]
Solving for \( x \):
\[ 2\left(1 + \frac{4}{\pi}\right)x = \frac{56}{\pi} \]
\[ x = \frac{28}{1 + \frac{4}{\pi}} = \frac{28\pi}{\pi + 4} \]

### Step 8: Find R
Substituting \( x \) back to find \( R \):
\[ R = \frac{14 - 2x}{\pi} = \frac{14 - 2\left(\frac{28\pi}{\pi + 4}\right)}{\pi} \]
Simplifying:
\[ R = \frac{14(\pi + 4) - 56\pi}{\pi(\pi + 4)} = \frac{14\cdot4 - 42\pi}{\pi + 4} = \frac{56 - 42\pi}{\pi + 4} \]

### Step 9: Calculate Lengths of Wire Pieces
The lengths of the wire pieces are:
- Length for square: \( 4x = \frac{112\pi}{\pi + 4} \)
- Length for circle: \( 2\pi R = \frac{28\pi}{\pi + 4} \)

### Final Answer
The lengths of the two pieces of wire should be:
- Length for square: \( \frac{112}{\pi + 4} \) meters
- Length for circle: \( \frac{28\pi}{\pi + 4} \) meters