To find the amount of fabric Krubell will need to make the pencil case in the shape of a circular cylinder, we need to calculate the surface area of the cylinder.
The surface area \( A \) of a cylinder is given by the formula:
\[ A = 2\pi rh + 2\pi r^2 \]
Where:
- \( r \) is the radius of the cylinder
- \( h \) is the height of the cylinder
In this case, we have:
- Radius \( r = 3 \) cm
- Height \( h = 22 \) cm
- Using \( \pi = 3.14 \)
Now, let's calculate the surface area step by step.
Step 1: Calculate the lateral surface area
The lateral surface area (the curved surface area) is given by:
\[ A_{\text{lateral}} = 2\pi rh = 2 \times 3.14 \times 3 \times 22 \]
Calculating this:
\[ A_{\text{lateral}} = 2 \times 3.14 \times 3 \times 22 = 2 \times 3.14 \times 66 = 2 \times 207.24 = 414.48 \text{ cm}^2 \]
Step 2: Calculate the area of the two circular bases
The area of one circular base is given by:
\[ A_{\text{base}} = \pi r^2 = 3.14 \times (3)^2 = 3.14 \times 9 = 28.26 \text{ cm}^2 \]
Since there are two bases:
\[ A_{\text{bases}} = 2 \times A_{\text{base}} = 2 \times 28.26 = 56.52 \text{ cm}^2 \]
Step 3: Calculate the total surface area
Now add the lateral surface area and the area of the bases:
\[ A = A_{\text{lateral}} + A_{\text{bases}} = 414.48 + 56.52 = 471 \text{ cm}^2 \]
Thus, the total amount of fabric Krubell will need to make the pencil case is:
\[ \boxed{471 \text{ cm}^2} \]
This matches option C: C. 471 cm²