Without knowing the size of the outer circle, it is impossible to determine the exact area of the shaded region. However, we can use the circumference of the inner circle (125.6 inches) to find its radius, and then use that to calculate the area of the shaded region as a fraction of the area of the outer circle.
The formula for the circumference of a circle is C = 2πr, where r is the radius. We can rearrange this formula to solve for r:
r = C/2π
Plugging in the given circumference of the inner circle, we get:
r = 125.6/2π
r ≈ 19.998 inches
Since both circles have the same center, we know that the radius of the outer circle must be at least 19.998 inches longer than the radius of the inner circle. Let's call the radius of the outer circle R. Then:
R = r + 19.998
R ≈ 39.996 inches
The area of a circle is given by the formula A = πr^2. So the area of the inner circle is:
A_inner = πr^2
A_inner ≈ 1256.64 square inches
And the area of the outer circle is:
A_outer = πR^2
A_outer ≈ 5023.27 square inches
The area of the shaded region is the difference between these two areas:
A_shaded = A_outer - A_inner
A_shaded ≈ 3766.63 square inches
So the area of the shaded region is approximately 3766.63 square inches, but this answer depends on the radius of the outer circle, which is not given in the problem.