Answer:
10053 cm²
Step-by-step explanation:
To find the total surface area of the dome made up of a combination of a hemisphere and a cylinder, we need to calculate the surface areas of each component and then sum them up.
Let's denote the radius of the hemisphere and the cylinder as [tex]\bold{\sf (r) }[/tex] (which is given as 20 cm).
Surface Area of the Hemisphere:
The surface area of a hemisphere is given by the formula:
[tex]\sf A_{\textsf{hemisphere}} = 2\pi r^2 [/tex]
[tex]\sf A_{\textsf{hemisphere}} = 2\pi (20 \, \textsf{cm})^2 [/tex]
Surface Area of the Cylinder:
The lateral surface area (LSA) of a cylinder is given by the formula:
[tex]\sf A_{\textsf{cylinder}} = 2\pi rh [/tex]
where
- [tex]\bold{\sf r }[/tex] is the radius of the base of the cylinder and
- [tex]\bold{\sf h }[/tex] is the height.
[tex]\sf A_{\textsf{cylinder}} = 2\pi (20 \, \textsf{cm}) (50 \, \textsf{cm}) [/tex]
Surface Area of the Circle (Base of the Cylinder):
The base of the cylinder is a circle, so its area is given by:
[tex]\sf A_{\textsf{circle}} = \pi r^2 [/tex]
[tex]\sf A_{\textsf{circle}} = \pi (20 \, \textsf{cm})^2 [/tex]
Total Surface Area:
Now, sum up the surface areas of the hemisphere, cylinder (including its base), and the base of the cylinder:
[tex]\sf A_{\textsf{total}} = A_{\textsf{hemisphere}} + A_{\textsf{cylinder}} + A_{\textsf{circle}} [/tex]
Calculate the numerical values.
[tex]\sf A_{\textsf{total}} = 2\pi (20 \, \textsf{cm})^2 + 2\pi (20 \, \textsf{cm}) (50 \, \textsf{cm}) + \pi (20 \, \textsf{cm})^2 [/tex]
[tex]\sf A_{\textsf{total}} = 800 \pi\, \textsf{cm}^2 + 2000\pi \, \textsf{cm}^2: + 400 \pi \, \textsf{cm}^2 [/tex]
[tex]\sf A_{\textsf{total}} = 3200 \pi\, \textsf{cm}^2 [/tex]
[tex]\sf A_{\textsf{total}} = 10053.096491487\textsf{cm}^2[/tex]
[tex]\sf A_{\textsf{total}} \approx 10053 \textsf{cm}^2 \, \textsf{(in nearest square centimeters)} [/tex]
Therefore, the total surface area of a dome is approximately 10053 cm².
