Question

The dome of the shape to the right is a hemisphere, whose
radius equals the radius of the right cylindrical base beneath
it. Find the total surface area, rounded to the nearest square
centimeter.

Answer 1

Answer:

10,053 cm²

Step-by-step explanation:

The total surface area of the given figure is made up of the curved surface of a hemisphere, the curved surface area of a cylinder, and the area of one circular base of the cylinder.

The curved surface area of a hemisphere is given by the formula:

$\text{Curved S.A. of a hemisphere}=2\pi r^2$

The curved surface area of a cylinder is called the lateral surface area (L.S.A.), and is given by the formula:

$\text{L.S.A. of a cylinder}=2\pi r h$

The area of a circle is given by the formula:

$\text{Area of a circle}=\pi r^2$

Therefore, the formula for the total surface area of the figure is:

$\begin{aligned}\text{Total S.A.}&=\text{Hemisphere}+\text{L.S.A. cylinder}+\text{circle}\\\\&=2\pi r^2 + 2\pi r h + \pi r^2\\\\&=3\pi r^2 + 2\pi r h\end{aligned}$

Given that the radius of the figure is 20 cm and the height of the cylinder is 50 cm, we can substitute r = 20 and h = 50 into the formula to find the total surface area of the figure:

$\begin{aligned}\text{Total S.A.}&=3\pi (20)^2 + 2\pi (20)(50)\\\\&=3\pi (400) + 2\pi (1000)\\\\&=1200\pi + 2000\pi\\\\&=3200\pi\\\\&=10053.09649...\\\\&=10053\; \sf cm^2\; (nearest\;square\;cm)\end{aligned}$

Therefore, the total surface area of the given figure is 10,053 cm² (rounded to the nearest square centimeter).

Answer 2

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 $\bold{\sf (r) }$ (which is given as 20 cm).

Surface Area of the Hemisphere:

The surface area of a hemisphere is given by the formula:

$\sf A_{\textsf{hemisphere}} = 2\pi r^2$

$\sf A_{\textsf{hemisphere}} = 2\pi (20 \, \textsf{cm})^2$

Surface Area of the Cylinder:

The lateral surface area (LSA) of a cylinder is given by the formula:

$\sf A_{\textsf{cylinder}} = 2\pi rh$

where

• $\bold{\sf r }$ is the radius of the base of the cylinder and
• $\bold{\sf h }$ is the height.

$\sf A_{\textsf{cylinder}} = 2\pi (20 \, \textsf{cm}) (50 \, \textsf{cm})$

Surface Area of the Circle (Base of the Cylinder):

The base of the cylinder is a circle, so its area is given by:

$\sf A_{\textsf{circle}} = \pi r^2$

$\sf A_{\textsf{circle}} = \pi (20 \, \textsf{cm})^2$

Total Surface Area:

Now, sum up the surface areas of the hemisphere, cylinder (including its base), and the base of the cylinder:

$\sf A_{\textsf{total}} = A_{\textsf{hemisphere}} + A_{\textsf{cylinder}} + A_{\textsf{circle}}$

Calculate the numerical values.

$\sf A_{\textsf{total}} = 2\pi (20 \, \textsf{cm})^2 + 2\pi (20 \, \textsf{cm}) (50 \, \textsf{cm}) + \pi (20 \, \textsf{cm})^2$

$\sf A_{\textsf{total}} = 800 \pi\, \textsf{cm}^2 + 2000\pi \, \textsf{cm}^2: + 400 \pi \, \textsf{cm}^2$

$\sf A_{\textsf{total}} = 3200 \pi\, \textsf{cm}^2$

$\sf A_{\textsf{total}} = 10053.096491487\textsf{cm}^2$

$\sf A_{\textsf{total}} \approx 10053 \textsf{cm}^2 \, \textsf{(in nearest square centimeters)}$

Therefore, the total surface area of a dome is approximately 10053 cm².