The sum of the radii of two circles is 91 centimetres, and the difference between their areas is 2002 square centimetres. Using pi = 22 / 7, what is the radius, in centimetres, of the larger circle?

Introduction / Context:
This circle problem connects the radii of two circles with the difference in their areas. We are told the sum of the radii and the difference in areas and asked to find the radius of the larger circle. The question is a straightforward application of circle area formula combined with algebraic factorisation and the idea that R^2 − r^2 can be written as (R − r)(R + r).

Given Data / Assumptions:

• Let R be the radius of the larger circle and r be the radius of the smaller circle.
• R + r = 91 cm.
• Difference of areas = pi * R^2 − pi * r^2 = 2002 cm^2.
• pi is to be taken as 22 / 7.
• We are required to find R, the radius of the larger circle.

Concept / Approach:
The area of a circle of radius x is pi * x^2. The difference of the areas of two circles with radii R and r is pi * (R^2 − r^2). Using the identity R^2 − r^2 = (R + r)(R − r), we can use the given sum R + r and area difference to determine R − r. Once both R + r and R − r are known, solving for R and r is a matter of solving a simple system of linear equations.

Step-by-Step Solution:
Step 1: Write the difference of areas using the formula for circle area: difference = pi * (R^2 − r^2) = 2002. Step 2: Substitute pi = 22 / 7 to get (22 / 7) * (R^2 − r^2) = 2002. Step 3: Solve for R^2 − r^2: R^2 − r^2 = 2002 * (7 / 22). Step 4: Compute 2002 / 22 = 91, so R^2 − r^2 = 91 * 7 = 637. Step 5: Use the identity R^2 − r^2 = (R + r)(R − r). We know R + r = 91, so (R + r)(R − r) = 637. Step 6: Substitute R + r = 91 to get 91 * (R − r) = 637. Step 7: Solve for R − r: R − r = 637 / 91 = 7. Step 8: Now we have two linear equations: R + r = 91 and R − r = 7. Step 9: Add the equations: (R + r) + (R − r) = 91 + 7, giving 2R = 98. Step 10: So R = 98 / 2 = 49 cm. The smaller radius would be r = 91 − 49 = 42 cm.

Verification / Alternative check:
To check, compute the areas with R = 49 and r = 42 using pi = 22 / 7. Area of larger circle = (22 / 7) * 49^2 = (22 / 7) * 2401. Since 2401 / 7 = 343, this gives 22 * 343 = 7546 cm^2. Area of smaller circle = (22 / 7) * 42^2 = (22 / 7) * 1764 = 22 * 252 = 5544 cm^2. The difference is 7546 − 5544 = 2002 cm^2, which matches the given value. This confirms that R = 49 cm is correct.

Why Other Options Are Wrong:
Choosing 56, 42, 63 or 35 for the larger radius leads either to a wrong sum of radii or to a difference of areas that is not equal to 2002 when computed using pi = 22 / 7. For example, if R = 56, then r = 35 to keep the sum 91, but the resulting area difference is different from 2002. Only R = 49 works consistently with both given conditions.

Common Pitfalls:
Common errors include forgetting to apply the factor pi on both circles, not using the difference of squares identity, or using only R^2 − r^2 directly without relating it to the given sum of radii. Some students also make mistakes simplifying 2002 * (7 / 22). Keeping the symbolic form pi * (R^2 − r^2) and carefully isolating R^2 − r^2 prevents such errors and leads naturally to the system R + r and R − r.

Final Answer:
The radius of the larger circle is 49 cm.