Two concentric circles with radii p cm and (p+ 2) cm are drawn on a paper. The difference between their areas is 44 sq. cm What is the value of p? (Take `pi=(22)/(7)`)

Options

A. 1.5
B. 5
C. 6
D. 2.5

Answer

To solve the problem, we need to find the value of \( p \) given the difference in the areas of two concentric circles. Let's break it down step by step.

### Step 1: Understand the areas of the circles
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
For the first circle with radius \( p \):
\[ A_1 = \pi p^2 \]
For the second circle with radius \( p + 2 \):
\[ A_2 = \pi (p + 2)^2 \]

### Step 2: Calculate the area of the second circle
Using the expansion formula for \( (a + b)^2 \):
\[ (p + 2)^2 = p^2 + 4p + 4 \]
Thus, the area of the second circle becomes:
\[ A_2 = \pi (p^2 + 4p + 4) = \pi p^2 + 4\pi p + 4\pi \]

### Step 3: Find the difference in areas
The difference in areas between the second and first circle is given by:
\[ A_2 - A_1 = ( \pi p^2 + 4\pi p + 4\pi ) - \pi p^2 \]
This simplifies to:
\[ A_2 - A_1 = 4\pi p + 4\pi \]

### Step 4: Set up the equation
According to the problem, the difference in areas is 44 sq. cm:
\[ 4\pi p + 4\pi = 44 \]

### Step 5: Factor out the common term
We can factor out \( 4\pi \):
\[ 4\pi (p + 1) = 44 \]

### Step 6: Solve for \( p + 1 \)
Dividing both sides by \( 4\pi \):
\[ p + 1 = \frac{44}{4\pi} = \frac{11}{\pi} \]

### Step 7: Substitute the value of \( \pi \)
Given \( \pi = \frac{22}{7} \):
\[ p + 1 = \frac{11}{\frac{22}{7}} = \frac{11 \times 7}{22} = \frac{77}{22} = \frac{7}{2} \]

### Step 8: Solve for \( p \)
Subtracting 1 from both sides:
\[ p = \frac{7}{2} - 1 = \frac{7}{2} - \frac{2}{2} = \frac{5}{2} \]
Thus, the value of \( p \) is:
\[ p = 2.5 \]

### Final Answer
The value of \( p \) is \( 2.5 \).

---