To find the total surface area of a soup can with a height of 4 inches and a radius of 1.25 inches, we need to consider both the area of the two circular bases and the area of the curved side surface of the cylinder.
1. Calculate the surface area of the two circular bases:
- The area of a single circular base is given by the formula:
[tex]\[
\text{Area of one base} = \pi \times \text{radius}^2
\][/tex]
- Given the radius is 1.25 inches, we can calculate the area of one base as:
[tex]\[
\text{Area of one base} = \pi \times (1.25)^2
\][/tex]
- The area of both circular bases combined is:
[tex]\[
\text{Total area of two bases} = 2 \times (\pi \times (1.25)^2)
\][/tex]
- Based on our calculation, the total area of the two circular bases is:
[tex]\[
9.817477042468104 \text{ square inches}
\][/tex]
2. Calculate the surface area of the rectangular side (cylindrical side):
- The side surface area of the cylinder is essentially a rectangle that wraps around the circular base. The height of this rectangle is the height of the can, and the width is the circumference of the circular base.
- The circumference of the circular base is given by:
[tex]\[
\text{Circumference} = 2 \times \pi \times \text{radius}
\][/tex]
- For a radius of 1.25 inches:
[tex]\[
\text{Circumference} = 2 \times \pi \times 1.25
\][/tex]
- The height of this rectangle is given as 4 inches.
- Therefore, the side surface area is:
[tex]\[
\text{Side surface area} = \text{Circumference} \times \text{Height}
\][/tex]
- Substituting the values gives:
[tex]\[
\text{Side surface area} = (2 \times \pi \times 1.25) \times 4
\][/tex]
- Based on our calculation, the side surface area is:
[tex]\[
31.41592653589793 \text{ square inches}
\][/tex]
3. Calculate the total surface area:
- The total surface area is the sum of the area of the two bases and the side surface area:
[tex]\[
\text{Total surface area} = \text{Total area of two bases} + \text{Side surface area}
\][/tex]
- Substituting the calculated values:
[tex]\[
\text{Total surface area} = 9.817477042468104 + 31.41592653589793
\][/tex]
- Therefore, the total surface area of the soup can is:
[tex]\[
41.23340357836604 \text{ square inches}
\][/tex]
So, the total surface area of the soup can is [tex]\( 41.23340357836604 \)[/tex] square inches.
