Three concentric circles have radii x cm, 10 cm and 5 cm such that x > 10 > 5. If the area enclosed by circles with radii x cm and 10 cm is the same as the area enclosed by the circles with radii 10 cm and 5 cm, find the area of the largest circle.

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Understand the Areas
We have three concentric circles with radii \( x \) cm, \( 10 \) cm, and \( 5 \) cm. The area enclosed by the circles with radii \( x \) cm and \( 10 \) cm is equal to the area enclosed by the circles with radii \( 10 \) cm and \( 5 \) cm.

### Step 2: Write the Area Formulas
The area of a circle is given by the formula \( A = \pi r^2 \). Therefore, the area enclosed by the circles with radii \( x \) cm and \( 10 \) cm is:
\[ \text{Area}_{x, 10} = \pi x^2 - \pi (10^2) = \pi (x^2 - 100) \]
The area enclosed by the circles with radii \( 10 \) cm and \( 5 \) cm is:
\[ \text{Area}_{10, 5} = \pi (10^2) - \pi (5^2) = \pi (100 - 25) = \pi (75) \]

### Step 3: Set the Areas Equal
Since the areas are equal, we can set up the equation:
\[ \pi (x^2 - 100) = \pi (75) \]

### Step 4: Simplify the Equation
We can divide both sides of the equation by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ x^2 - 100 = 75 \]

### Step 5: Solve for \( x^2 \)
Now, we can solve for \( x^2 \):
\[ x^2 = 75 + 100 \]
\[ x^2 = 175 \]

### Step 6: Find \( x \)
Taking the square root of both sides gives us:
\[ x = \sqrt{175} \]
Calculating this gives:
\[ x \approx 13.23 \text{ cm} \]

### Step 7: Calculate the Area of the Largest Circle
Now, we need to find the area of the largest circle, which has radius \( x \):
\[ \text{Area}_{\text{largest}} = \pi x^2 = \pi (175) \]
Using \( \pi \approx \frac{22}{7} \):
\[ \text{Area}_{\text{largest}} = \frac{22}{7} \times 175 \]
Calculating this gives:
\[ \text{Area}_{\text{largest}} = 550 \text{ cm}^2 \]

### Final Answer
The area of the largest circle is \( 550 \text{ cm}^2 \).

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