To determine the area of the paper that will be used to make the label of the soup can, we need to find the lateral surface area of the cylinder. Let's go through the steps to achieve this:
1. Convert the diameter to a radius:
The diameter of the can is given as [tex]\(2 \frac{3}{8}\)[/tex] inches. First, convert this mixed number to an improper fraction:
[tex]\[
2 \frac{3}{8} = 2 + \frac{3}{8} = \frac{16}{8} + \frac{3}{8} = \frac{19}{8} \text{ inches}
\][/tex]
The radius [tex]\(r\)[/tex] is half of the diameter:
[tex]\[
r = \frac{\frac{19}{8}}{2} = \frac{19}{16} \text{ inches}
\][/tex]
2. Calculate the circumference of the base:
The circumference [tex]\(C\)[/tex] of a circle is given by the formula [tex]\(C = 2\pi r\)[/tex]. Using [tex]\( \pi \approx 3.14159 \)[/tex] and [tex]\( r = \frac{19}{16} \)[/tex]:
[tex]\[
C = 2 \times 3.14159 \times \frac{19}{16} \approx 2 \times 3.14159 \times 1.1875 \approx 2 \times 3.14159 \times 1.1875 \approx 2 \times 3.730639375 \approx 7.46127875 \text{ inches}
\][/tex]
3. Calculate the lateral surface area:
The lateral surface area [tex]\(A\)[/tex] of a cylinder is given by the formula [tex]\(A = \text{circumference} \times \text{height}\)[/tex]. Given the height [tex]\(h = 5\)[/tex] inches:
[tex]\[
A = 7.46127875 \times 5 \approx 37.30639375 \text{ square inches}
\][/tex]
4. Round the lateral surface area to two decimal places:
The rounded lateral surface area is:
[tex]\[
A \approx 37.31 \text{ square inches}
\][/tex]
Therefore, the area of the paper that will be used to make the label of the can is [tex]\( \boxed{37.31} \)[/tex] square inches.
