If the radius ( r ) of a circle is increased by 'x' units , what is the number of units by which the circumference of the circle is increased?
A. `pi `
B. ` 2pi `
C. ` 2pi r `
D. ` 2pi x `

Options

A. D
B. C
C. B
D. A

Answer

To find the increase in the circumference of a circle when the radius is increased by 'x' units, we can follow these steps:

### Step 1: Understand the initial radius and circumference
Let the initial radius of the circle be \( r \). The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]

### Step 2: Determine the new radius
If the radius is increased by \( x \) units, the new radius \( R \) becomes:
\[ R = r + x \]

### Step 3: Calculate the new circumference
Using the new radius, the new circumference \( C' \) can be calculated as:
\[ C' = 2\pi R = 2\pi (r + x) = 2\pi r + 2\pi x \]

### Step 4: Find the increase in circumference
The increase in circumference \( \Delta C \) can be found by subtracting the initial circumference from the new circumference:
\[ \Delta C = C' - C = (2\pi r + 2\pi x) - 2\pi r \]
\[ \Delta C = 2\pi x \]

### Conclusion
Thus, the increase in the circumference of the circle when the radius is increased by \( x \) units is:
\[ \Delta C = 2\pi x \]

### Answer
The correct option is **D. \( 2\pi x \)**.

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