First, let's divide our solid into two parts:
1. A parallelepiped without the top face
The faces of this portion of the solid would be:
• 2, square faces measuring 4cm x 4cm each
,
• 3 ,rectangular faces measuring 4cm x 20cm each
Therefore, the surface area of this portion would be:
[tex]2(4cm\cdot4cm)+3(4cm\cdot20cm)=272\operatorname{cm}^2[/tex]
2. Half a cylinder
We know that the surface area of a cylinder is:
[tex]2\pi rh+2\pi r^2[/tex]
If we divide it by two, we'll get the surface area of the half cylinder we need:
[tex]\frac{2\pi rh+2\pi r^2}{2}\rightarrow\pi rh+\pi r^2[/tex]
Now, from the figure we can conclude that the diameter is 4 cm. Therefore, the radius would be 2 cm. Also, the height of this semicylinder is 20cm. Using this data, we'll have that the surface area for this part of the solid would be:
[tex]\begin{gathered} \pi rh+\pi r^2\rightarrow\pi\cdot(2cm)\cdot(20cm)+\pi(2\operatorname{cm})^2 \\ \Rightarrow138.23\operatorname{cm}^2 \end{gathered}[/tex]
Now, to get the total surface area of this solid, we'll need to add up the two results we just got:
[tex]272\operatorname{cm}+138.23\operatorname{cm}=410.23\operatorname{cm}^2[/tex]
Therefore, we can conclude that the surface area of the solid is:
[tex]410.23\operatorname{cm}^2[/tex]
