Sure, let’s work through this problem step-by-step to determine the length of the label:
1. Identify the radius: The radius of the soda can is given as [tex]\(4 \frac{1}{4} \)[/tex] cm. To make calculations easier, convert this mixed fraction into an improper fraction or a decimal.
[tex]\[
4 \frac{1}{4} = 4 + \frac{1}{4} = 4 + 0.25 = 4.25 \text{ cm}
\][/tex]
2. Circumference formula: To find the length of the label, we need the circumference of the soda can, which can be calculated using the formula:
[tex]\[
C = 2\pi r
\][/tex]
where [tex]\(C\)[/tex] is the circumference, [tex]\(\pi\)[/tex] is approximately [tex]\(\frac{22}{7}\)[/tex], and [tex]\(r\)[/tex] is the radius.
3. Substitute the values: Substitute [tex]\( r = 4.25 \)[/tex] cm and [tex]\(\pi = \frac{22}{7}\)[/tex] into the formula:
[tex]\[
C = 2 \times \frac{22}{7} \times 4.25
\][/tex]
4. Perform the multiplication:
[tex]\[
C = 2 \times \frac{22}{7} \times 4.25
\][/tex]
5. Multiply step-by-step:
[tex]\[
2 \times \frac{22}{7} = \frac{44}{7}
\][/tex]
[tex]\[
\frac{44}{7} \times 4.25 = \frac{44 \times 4.25}{7} = \frac{44 \times \frac{17}{4}}{7}
\][/tex]
because [tex]\(4.25 = \frac{17}{4}\)[/tex].
6. Simplify multiplication:
[tex]\[
\frac{44 \times 17}{4 \times 7} = \frac{748}{28}
\][/tex]
7. Final result: Simplify [tex]\(\frac{748}{28}\)[/tex]:
[tex]\[
\frac{748}{28} = 26.714285714285715
\][/tex]
8. Convert the numerical result back into one of the provided choices:
[tex]\[
26.714285714285715 = \frac{748}{28}
\][/tex]
Simplifying this fraction, we get:
[tex]\[
\frac{748}{28} = \frac{187}{7}
\][/tex]
Therefore, the correct length of the label, using [tex]\(\frac{22}{7}\)[/tex] for [tex]\(\pi\)[/tex], is:
[tex]\[
\boxed{\frac{187}{7} \text{ cm}}
\][/tex]
