The‌ ‌difference‌ ‌between‌ ‌the‌ ‌ circumference‌ ‌and‌ ‌the‌ ‌radius‌ ‌of‌ ‌the‌ ‌circle‌ ‌is‌ ‌37‌ ‌cm.‌ ‌The‌ ‌area‌ ‌of‌ ‌the‌ ‌circle‌ ‌will‌ ‌be‌

Options

A. `110cm^(2)`
B. `84cm^(2)`
C. `154cm^(2)`
D. `44 cm^(2)`

Answer

To solve the problem step by step, we need to find the area of a circle given that the difference between its circumference and radius is 37 cm.

### Step 1: Understand the given information
We know that:
- The difference between the circumference (C) and the radius (r) is 37 cm.
- This can be expressed as:
\[ C - r = 37 \]

### Step 2: Write the formula for circumference
The formula for the circumference of a circle is:
\[ C = 2\pi r \]
Substituting this into the equation from Step 1 gives:
\[ 2\pi r - r = 37 \]

### Step 3: Simplify the equation
We can factor out r from the left side:
\[ r(2\pi - 1) = 37 \]

### Step 4: Solve for r
To find the radius (r), we rearrange the equation:
\[ r = \frac{37}{2\pi - 1} \]
Now, substituting \(\pi\) with \(\frac{22}{7}\):
\[ r = \frac{37}{2 \times \frac{22}{7} - 1} \]

### Step 5: Calculate the value of r
Calculating the denominator:
\[ 2 \times \frac{22}{7} = \frac{44}{7} \]
So,
\[ 2\pi - 1 = \frac{44}{7} - 1 = \frac{44}{7} - \frac{7}{7} = \frac{37}{7} \]
Now substituting back:
\[ r = \frac{37}{\frac{37}{7}} = 37 \times \frac{7}{37} = 7 \text{ cm} \]

### Step 6: Find the area of the circle
Now that we have the radius, we can find the area (A) using the formula:
\[ A = \pi r^2 \]
Substituting the value of r:
\[ A = \frac{22}{7} \times (7^2) = \frac{22}{7} \times 49 \]
Calculating this gives:
\[ A = \frac{22 \times 49}{7} = 22 \times 7 = 154 \text{ cm}^2 \]

### Final Answer
The area of the circle is \(154 \text{ cm}^2\).

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