Let's work through the problem step-by-step to determine the surface area covered by the label on the cylindrical wheel of cheese.
### Step 1: Identify the Given Information
- Radius ([tex]\(r\)[/tex]) of the cylinder: [tex]\(22\)[/tex] cm
- Height ([tex]\(h\)[/tex]) of the cylinder: [tex]\(16\)[/tex] cm
- Value of [tex]\(\pi\)[/tex]: [tex]\(\frac{22}{7}\)[/tex]
### Step 2: Calculate the Circumference of the Base of the Cylinder
The circumference [tex]\(C\)[/tex] of the base of the cylinder can be calculated using the following formula:
[tex]\[ C = 2 \pi r \][/tex]
Substitute the given values:
[tex]\[ C = 2 \times \frac{22}{7} \times 22 \][/tex]
### Step 3: Calculate the Lateral Surface Area of the Cylinder
The lateral surface area [tex]\(A\)[/tex] of the cylinder, which is the area of the label, can be found using the formula:
[tex]\[ A = \text{circumference} \times \text{height} \][/tex]
Substitute the value of the circumference we calculated:
[tex]\[ A = \left(2 \times \frac{22}{7} \times 22\right) \times 16 \][/tex]
### Step 4: Interpret the Given Answer
Given the results:
- The circumference of the base of the cylinder is approximately [tex]\(138.28571428571428\)[/tex] cm.
- The lateral surface area of the cylinder is approximately [tex]\(2212.5714285714284\)[/tex] square centimeters.
### Step 5: Express the Lateral Surface Area in Terms of a Fraction
The given multiple-choice answers are all represented as fractions with a numerator over [tex]\(7\)[/tex]. We can convert the lateral surface area value to this form:
- Lateral surface area: [tex]\(2212.5714285714284\)[/tex]
- This can be rewritten approximately as [tex]\(\frac{15488}{7}\)[/tex] square centimeters.
Thus, the label covers approximately [tex]\(\frac{15488}{7}\)[/tex] square centimeters.
### Conclusion
The correct multiple-choice answer that matches our calculation is:
[tex]\[
\boxed{\frac{15,488}{7}}
\][/tex]
