Problem:
The area of a circular ring enclosed between two concentric circles is 286 cm². Find the radii of the two circles, given that their difference is 7 cm.

Solution:

Let's assume that the radius of the larger circle is 'R' and the radius of the smaller circle is 'r'. We are given that the difference between the two radii is 7 cm.

Step 1: Set up the equation
The area of the circular ring is given by the difference between the areas of the larger and smaller circles. The equation for the area of a circle is A = πr².

We can set up the equation as follows:
Area of circular ring = Area of larger circle - Area of smaller circle

286 cm² = πR² - πr²

Step 2: Simplify the equation
Since we know that the difference between the radii is 7 cm, we can express the larger radius 'R' in terms of the smaller radius 'r':

R = r + 7

Now, substitute the value of R in the equation:

286 cm² = π(r + 7)² - πr²

Expand the equation:

286 cm² = π(r² + 14r + 49) - πr²

Step 3: Solve the equation
Simplify the equation:

286 cm² = πr² + 14πr + 49π - πr²

286 cm² = 14πr + 49π

Divide both sides of the equation by π:

286 cm²/π = 14r + 49

Simplify further:

91.07 cm² ≈ 14r + 49

Subtract 49 from both sides:

91.07 cm² - 49 ≈ 14r

Simplify:

42.07 cm² ≈ 14r

Divide both sides by 14:

42.07 cm²/14 ≈ r

r ≈ 3.005 cm

Step 4: Calculate the radius of the larger circle
Since we know that the difference between the radii is 7 cm, we can calculate the radius of the larger circle 'R' using the equation:

R = r + 7

Substituting the value of r:

R = 3.005 cm + 7

R ≈ 10.005 cm

Step 5: Final answer
The radius of the smaller circle is approximately 3.005 cm, and the radius of the larger circle is approximately 10.005 cm.

R = 3 cm for smaller circle
and R = 10 cm for greater cicle