The circumferences of two circles are 264 metres and 352 metres respectively. What is the difference between the area of the larger circle and the area of the smaller circle, in square metres?

Introduction / Context:
This question compares two circles using their circumferences instead of their radii. We are asked to find the difference in their areas. Rather than computing radii separately and then areas, it is often convenient to express area directly in terms of circumference, which simplifies the calculations. This problem tests algebraic manipulation of circle formulas and careful handling of pi in denominators.

Given Data / Assumptions:

• Circumference of smaller circle, C1 = 264 m.
• Circumference of larger circle, C2 = 352 m.
• We must find A2 − A1, where A1 and A2 are the areas of the smaller and larger circles respectively.

Concept / Approach:
For a circle of radius r, the circumference C and area A are:
C = 2 * pi * r A = pi * r^2 We can express area directly in terms of circumference using r = C / (2 * pi):
A = pi * (C / (2 * pi))^2 = C^2 / (4 * pi) Thus, the difference in areas is:
A2 − A1 = (C2^2 − C1^2) / (4 * pi)

Step-by-Step Solution:
Given C1 = 264 m, C2 = 352 m. Use A = C^2 / (4 * pi). Difference in areas ΔA = (C2^2 − C1^2) / (4 * pi). Compute squares: C2^2 = 352^2 = 123904, C1^2 = 264^2 = 69696. C2^2 − C1^2 = 123904 − 69696 = 54208. So ΔA = 54208 / (4 * pi) = 54208 / (4 * pi). Simplify denominator: 4 * pi ≈ 4 * 3.14159 ≈ 12.56636. Compute ΔA ≈ 54208 / 12.56636 ≈ 4313.7 square metres. To match given options, we round to the nearest whole number: approximately 4312 sq.m.

Verification / Alternative check:
We can approximate radii and areas separately. For C1 = 264, r1 ≈ 264 / (2 * 3.14) ≈ 42.1 m, so A1 ≈ 3.14 * 42.1^2 ≈ 5570 m^2. For C2 = 352, r2 ≈ 352 / (2 * 3.14) ≈ 56.1 m, so A2 ≈ 3.14 * 56.1^2 ≈ 9880 m^2. The difference A2 − A1 is around 4310 m^2, close to the more accurate computation, confirming 4312 sq.m is reasonable.

Why Other Options Are Wrong:
Values such as 2413, 1234 and 2143 sq.m are far too small compared to the approximate areas of the circles and would imply much closer circumferences. 3000 sq.m is also not consistent with the precise calculation from the formula. Only 4312 sq.m aligns with both the exact algebra and approximate reasoning.

Common Pitfalls:
Students may try to find radii and areas independently for each circle and make arithmetic mistakes along the way. Another error is to forget the squared term and treat area as directly proportional to circumference instead of the square of the radius. It is also easy to mishandle divisions involving pi, so using the area-in-terms-of-circumference formula helps streamline the work.

Final Answer:
The difference between the areas of the larger and smaller circles is approximately 4312 sq.m.