The Problem:
We are given a circle and we need to find out what happens to its area when its radius is increased in such a way that its circumference increases by 15%.

Understanding the Problem:
To solve this problem, we need to know the formulas for calculating the circumference and area of a circle.

The circumference of a circle is given by the formula:
C = 2πr

The area of a circle is given by the formula:
A = πr^2

Solution:
Let's assume the initial radius of the circle is 'r'.

1. Calculate the initial circumference of the circle:
Initial Circumference (C1) = 2πr

2. Calculate the increase in circumference:
Increase in Circumference = 15% of C1 = 0.15 * C1

3. Calculate the new circumference of the circle:
New Circumference (C2) = C1 + Increase in Circumference = C1 + 0.15 * C1 = C1(1 + 0.15)

4. Calculate the new radius of the circle:
New Circumference (C2) = 2π * New Radius (R2)
New Radius (R2) = C2 / (2π) = (C1(1 + 0.15)) / (2π)

5. Calculate the area of the circle with the new radius:
New Area (A2) = π * R2^2 = π * ((C1(1 + 0.15)) / (2π))^2 = π * (C1(1 + 0.15))^2 / (4π^2)

6. Simplify the expression:
New Area (A2) = (C1(1 + 0.15))^2 / (4π)

7. Expand the expression:
New Area (A2) = (C1^2(1 + 0.15)^2) / (4π)

8. Simplify further:
New Area (A2) = (C1^2(1 + 0.3 + 0.0225)) / (4π)
New Area (A2) = (C1^2(1.3225)) / (4π)

9. Simplify the expression:
New Area (A2) = 1.3225 * (C1^2) / (4π)

10. Calculate the increase in area:
Increase in Area = New Area (A2) - Initial Area (A1) = (1.3225 * (C1^2) / (4π)) - (πr^2)

11. Simplify the expression:
Increase in Area = (1.3225 * (C1^2) - 4πr^2) / (4π)

12. Substitute the value of C1 and simplify further:
Increase in Area = (1.3225 * (2πr)^2 - 4πr^2) / (4π)
Increase in Area = (1.3225 * 4π^2r^2 - 4πr^2) / (4π)
Increase in Area = (5.29π^2r^2 - 4πr^2) / (4π)
Increase in Area = πr