Verified answer

Answer:

Let's denote the side length of the square as "x" units and the radius of the circle as "r" units.

1. Area of the square = x^2 square units

2. Area of the circle = πr^2 square units

The sum of the areas of the square and the circle is:

Sum = x^2 + πr^2

We need to find the values of "x" and "r" for which the sum of the areas is minimized.

Given that the wire's length is 2 units and is cut into two parts to form the square and the circle, we have the following relationship:

Perimeter of the square + Circumference of the circle = Length of the wire

4x + 2πr = 2

Now, we can solve this system of equations to find the values of "x" and "r."

From the second equation, we can isolate "r" in terms of "x":

2πr = 2 - 4x

r = (2 - 4x) / (2π)

Now, substitute this value of "r" into the area equation:

Sum = x^2 + π[(2 - 4x) / (2π)]^2

Sum = x^2 + (2 - 4x)^2 / (4π)

To minimize the sum, we find the critical point by setting the derivative of the sum with respect to "x" to zero:

d/dx [x^2 + (2 - 4x)^2 / (4π)] = 0

2x - (4/π) (2 - 4x) = 0

2x - (8/π) + (16/π) x = 0

(16/π) x + 2x = (8/π)

(16/π + 2) x = (8/π)

x = (8/π) / (16/π + 2)

x = 8 / (16 + 2π)

Now, to find the value of "r," substitute the value of "x" back into the expression we derived earlier for "r":

r = (2 - 4(8 / (16 + 2π))) / (2π)

r = (2 - (32 / (16 + 2π))) / (2π)

r = (2(16 + 2π) - 32) / (2π * (16 + 2π))

r = (32π + 4π^2 - 32) / (32π + 4π^2)

Now, let's simplify the expressions for "x" and "r":

x = 8 / (16 + 2π)

r = (32π + 4π^2 - 32) / (32π + 4π^2)

Now, let's check which option matches the simplified expressions for "x" and "r."

(4-π)x = (4 - π) * (8 / (16 + 2π)) = (32 - 8π) / (16 + 2π)

πr = π * [(32π + 4π^2 - 32) / (32π + 4π^2)] = (32π^2 + 4π^3 - 32π) / (32π + 4π^2)

As we can see, the expressions (4-π)x and πr are not equal. So, option (1) is not correct.

Next, let's check option (2):

x = 8 / (16 + 2π)

2r = 2 * [(32π + 4π^2 - 32) / (32π + 4π^2)] = (64π + 8π^2 - 64) / (32π + 4π^2) = (8π^2) / (8π^2) = 1

The values of "x" and "r" in option (2) are not equal, so option (2) is not correct.

Now, let's check option (3):

2x = 2 * (8 / (16 + 2π)) = 16 / (16 + 2π)

r = (32π + 4π^2 - 32) / (32π + 4π^2)

The values of "2x" and "r" are not equal, so option (3) is not correct.

Finally, let's check option (4):

2x = 2 * (8 / (16 + 2π)) = 16 / (16 + 2π)

(π + 4) * r = (π + 4) * [(32π + 4π^2 - 32) / (32π + 4π^2)] = [(32π^2 + 4π^3 - 32π + 128π + 16π^2 - 128) / (32π + 4π^2)]

= (16π^2 + 4π^3) / (32π + 4π^2)

The values of "2x" and "(π + 4) * r" are not equal, so option (4) is also not correct.

After checking all the options, we can conclude that none of the provided options is correct. The expressions for "x" and "r" obtained above cannot be simplified further into any of the given options.

Step-by-step explanation:

Let's denote the side of the square as 'x' units and the radius of the circle as 'r' units.

1. Area of the square = x^2 square units

2. Area of the circle = πr^2 square units

The sum of the areas of the square and the circle is given by:

Sum = x^2 + πr^2

Since the wire's total length is 2 units, it is used to form both the square and the circle:

Perimeter of the square = 4x units

Circumference of the circle = 2πr units

Perimeter of the square + Circumference of the circle = 2 units

4x + 2πr = 2

Now, we need to express 'r' in terms of 'x' to minimize the sum of the areas:

1. Solve for 'r' in the equation: 4x + 2πr = 2

  2πr = 2 - 4x

  r = (2 - 4x) / (2π)

2. Substitute the value of 'r' into the sum of areas equation:

  Sum = x^2 + π((2 - 4x) / (2π))^2

  Simplify:

  Sum = x^2 + (1/4)(2 - 4x)^2

  Sum = x^2 + (1/4)(4 - 16x + 16x^2)

  Sum = x^2 + (1/4)(16x^2 - 16x + 4)

  Sum = x^2 + 4x^2 - 4x + 1

  Sum = 5x^2 - 4x + 1

To find the minimum value of the sum, we differentiate it with respect to 'x' and set the derivative to zero:

d(Sum)/dx = 10x - 4

Now, set the derivative to zero and solve for 'x':

10x - 4 = 0

10x = 4

x = 4/10

x = 2/5

To find the corresponding value of 'r', substitute the value of 'x' back into the expression for 'r':

r = (2 - 4(2/5)) / (2π)

r = (2 - 8/5) / (2π)

r = (10/5 - 8/5) / (2π)

r = 2/5π

So, the values of 'x' and 'r' that minimize the sum of the areas are x = 2/5 and r = 2/5π.

Now, let's check the answer options:

(1) (4-π)x = πr

  (4-π)(2/5) = π(2/5π)

  (8/5 - 2π/5) = 2/5 (Not true)

(2) x = 2r

  2/5 = 2(2/5π)

  2/5 = 4/5π (Not true)

(3) 2x = r

  2(2/5) = 2/5π (Not true)

(4) 2x = (π + 4)r

  2(2/5) = (π + 4)(2/5π)

  4/5 = 2/5 + 8/5 (Not true)

None of the answer options match the condition for the minimum sum of the areas. Therefore, the correct answer is not listed in the options. The correct values for x and r that minimize the sum of the areas are x = 2/5 and r = 2/5π.