Sum of Area of circle and a rectangle is equal to 622 sq cm. The diameter of the circle is 14 cm, then what is the sum of circumference of the circle and the perimeter of the rectangle if the length of rectangle is 26 cm?

Options

A. A)142cm
B. B)132cm
C. C)152cm
D. D)140cm

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the area of the circle.
The formula for the area of a circle is given by:
\[ \text{Area of Circle} = \pi r^2 \]
Where \( r \) is the radius of the circle. The diameter of the circle is given as 14 cm, so the radius \( r \) is:
\[ r = \frac{\text{Diameter}}{2} = \frac{14}{2} = 7 \text{ cm} \]
Now, substituting the radius into the area formula:
\[ \text{Area of Circle} = \pi (7^2) = \pi \times 49 \]
Using \( \pi \approx \frac{22}{7} \):
\[ \text{Area of Circle} = \frac{22}{7} \times 49 = \frac{22 \times 49}{7} = 22 \times 7 = 154 \text{ sq cm} \]

### Step 2: Set up the equation for the area of the rectangle.
We know that:
\[ \text{Area of Circle} + \text{Area of Rectangle} = 622 \text{ sq cm} \]
Substituting the area of the circle:
\[ 154 + \text{Area of Rectangle} = 622 \]
Now, solving for the area of the rectangle:
\[ \text{Area of Rectangle} = 622 - 154 = 468 \text{ sq cm} \]

### Step 3: Calculate the width of the rectangle.
The area of a rectangle is given by:
\[ \text{Area of Rectangle} = \text{Length} \times \text{Width} \]
The length of the rectangle is given as 26 cm, so we can set up the equation:
\[ 468 = 26 \times \text{Width} \]
Now, solving for the width:
\[ \text{Width} = \frac{468}{26} = 18 \text{ cm} \]

### Step 4: Calculate the circumference of the circle.
The formula for the circumference of a circle is given by:
\[ \text{Circumference} = \pi d \]
Where \( d \) is the diameter. Substituting the diameter:
\[ \text{Circumference} = \pi \times 14 = \frac{22}{7} \times 14 \]
Calculating this:
\[ \text{Circumference} = \frac{22 \times 14}{7} = 22 \times 2 = 44 \text{ cm} \]

### Step 5: Calculate the perimeter of the rectangle.
The formula for the perimeter of a rectangle is given by:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Substituting the values:
\[ \text{Perimeter} = 2 \times (26 + 18) = 2 \times 44 = 88 \text{ cm} \]

### Step 6: Calculate the sum of the circumference of the circle and the perimeter of the rectangle.
Now we add the circumference of the circle and the perimeter of the rectangle:
\[ \text{Sum} = \text{Circumference} + \text{Perimeter} = 44 + 88 = 132 \text{ cm} \]

### Final Answer:
The sum of the circumference of the circle and the perimeter of the rectangle is **132 cm**.

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