Answer:
[tex]S\!A=52.5\pi \text{ cm}^2[/tex]
[tex]S\!A\approx164.93 \text{ cm}^2[/tex]
Step-by-step explanation:
We can solve for the surface area of a cylinder (in this case, a can of soup) by adding the areas of each of its sides.
A cylinder is composed of:
- 2 bases, which are both circles
- 1 curved side, which is a rectangle
The areas of these two parts, respectively, is:
- [tex]A_\circ = \pi r^2[/tex]
- [tex]A_\square = w \cdot h[/tex]
And, we know that the width of the rectangular side is the same as the circumference of the base circle, so the area of the side becomes:
- [tex]A_\text{side} = 2\pi r \cdot h[/tex]
Now, we can solve for the total surface area by adding up the expressions for the areas of the sides:
[tex]S\!A = A_\circ + A_\circ + A_\text{side}[/tex]
↓ combining like terms
[tex]S\!A = 2(A_\circ) + A_\text{side}[/tex]
↓ substituting in the known expressions
[tex]S\!A = 2(\pi r^2) + 2\pi r \cdot h[/tex]
↓ plugging in the given dimensions
[tex]S\!A = 2\pi(2.5^2) + 2\pi(2.5)(8)[/tex]
↓ executing the multiplication
[tex]S\!A = 12.5\pi + 40\pi[/tex]
↓ combining like terms
[tex]S\!A=52.5\pi \text{ cm}^2[/tex]
↓ approximating using a calculator
[tex]\boxed{S\!A\approx164.93 \text{ cm}^2}[/tex]
