Area and perimeter of a rectangle is `432 cm^(2)` and 84 cm respectively. If the rectangle is inscribed in a circle of maximum possible area, then find the circumference of the circle.

Options

A. A)`72 (4)/(7) cm`
B. B)`68 (3)/(7) cm`
C. C)`56 (5)/(7) cm`
D. D)`94 (2)/(7) cm`

Answer

To solve the problem, we need to find the circumference of a circle that inscribes a rectangle with a given area and perimeter. Let's break this down step by step.

### Step 1: Set up the equations
We know that:
- The area of the rectangle (A) = length (l) × width (w) = 432 cm²
- The perimeter of the rectangle (P) = 2(l + w) = 84 cm

### Step 2: Simplify the perimeter equation
From the perimeter equation, we can simplify:
\[ l + w = \frac{P}{2} = \frac{84}{2} = 42 \text{ cm} \]

### Step 3: Set up a system of equations
Now we have two equations:
1. \( lw = 432 \) (1)
2. \( l + w = 42 \) (2)

### Step 4: Express one variable in terms of the other
From equation (2), we can express \( w \) in terms of \( l \):
\[ w = 42 - l \]

### Step 5: Substitute into the area equation
Substituting \( w \) in equation (1):
\[ l(42 - l) = 432 \]
Expanding this gives:
\[ 42l - l^2 = 432 \]
Rearranging gives:
\[ l^2 - 42l + 432 = 0 \]

### Step 6: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[ l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -42, c = 432 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-42)^2 - 4(1)(432) = 1764 - 1728 = 36 \]

Now substituting back into the formula:
\[ l = \frac{42 \pm \sqrt{36}}{2} = \frac{42 \pm 6}{2} \]

Calculating the two possible values for \( l \):
1. \( l = \frac{48}{2} = 24 \)
2. \( l = \frac{36}{2} = 18 \)

### Step 7: Find the dimensions of the rectangle
Thus, the dimensions of the rectangle are:
- Length \( l = 24 \) cm
- Width \( w = 18 \) cm

### Step 8: Calculate the diagonal of the rectangle
The diagonal \( d \) of the rectangle can be calculated using the Pythagorean theorem:
\[ d = \sqrt{l^2 + w^2} = \sqrt{24^2 + 18^2} = \sqrt{576 + 324} = \sqrt{900} = 30 \text{ cm} \]

### Step 9: Determine the diameter of the circle
Since the rectangle is inscribed in the circle, the diameter of the circle is equal to the diagonal of the rectangle:
\[ \text{Diameter of the circle} = d = 30 \text{ cm} \]

### Step 10: Calculate the circumference of the circle
The circumference \( C \) of the circle is given by:
\[ C = \pi \times d = \pi \times 30 \]

Using \( \pi \approx \frac{22}{7} \):
\[ C = \frac{22}{7} \times 30 = \frac{660}{7} \text{ cm} \]

Converting this to a mixed number:
\[ C = 94 \frac{2}{7} \text{ cm} \]

### Final Answer
The circumference of the circle is \( 94 \frac{2}{7} \text{ cm} \).

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