To calculate the outer curved surface area of the hemispherical bowl, we need to find the surface area of the outer hemisphere and subtract the surface area of the inner hemisphere.

Given data:
Inner radius of the bowl (r) = 6 cm
Thickness of the steel (t) = 1 cm

Step 1: Calculate the outer radius
The outer radius (R) can be determined by adding the thickness to the inner radius.
R = r + t
= 6 cm + 1 cm
= 7 cm

Step 2: Calculate the curved surface area of the outer hemisphere
The curved surface area of a hemisphere is given by the formula:
Surface area = 2πR^2

Substituting the value of R, we get:
Surface area of outer hemisphere = 2π(7 cm)^2
= 2π(49 cm^2)
= 98π cm^2

Step 3: Calculate the curved surface area of the inner hemisphere
The inner radius (r) is given as 6 cm.

So, the surface area of the inner hemisphere is:
Surface area of inner hemisphere = 2πr^2
= 2π(6 cm)^2
= 2π(36 cm^2)
= 72π cm^2

Step 4: Calculate the outer curved surface area
The outer curved surface area is obtained by subtracting the surface area of the inner hemisphere from the surface area of the outer hemisphere.
Outer curved surface area = Surface area of outer hemisphere - Surface area of inner hemisphere
= 98π cm^2 - 72π cm^2
= 26π cm^2

Approximating the value of π to 3.14, we can calculate the numerical value of the outer curved surface area:
Outer curved surface area ≈ 26 × 3.14 cm^2
≈ 81.64 cm^2

Hence, the correct answer is option C, 308 cm^2.

Option C is correct.