Given:
A diagram of a composite figure.
Radius of cone and hemisphere is 8 cm.
Height of the cone is 15 cm.
To find:
The volume and the surface area of the composite figure.
Solution:
Volume of a cone is:
[tex]V_1=\dfrac{1}{3}\pi r^2h[/tex]
Where, r is the radius and h is the height of the cone.
Putting [tex]r=8,h=15[/tex] in the above formula, we get
[tex]V_1=\dfrac{1}{3}\pi (8)^2(15)[/tex]
[tex]V_1=\pi (64)(5)[/tex]
[tex]V_1=320\pi[/tex]
Volume of the hemisphere is:
[tex]V_2=\dfrac{2}{3}\pi r^3[/tex]
Where, r is the radius.
Putting [tex]r=8[/tex], we get
[tex]V_2=\dfrac{2}{3}\pi (8)^3[/tex]
[tex]V_2=\dfrac{1024}{3}\pi [/tex]
[tex]V_2\approx 341.3\pi [/tex]
Now, the volume of the composite figure is:
[tex]V=V_1+V_2[/tex]
[tex]V=320\pi +341.3\pi[/tex]
[tex]V=661.3\pi[/tex]
The volume of the composite figure is 661.3π cm³.
The curved surface area of a cone is:
[tex]A_1=\pi r\sqrt{h^2+r^2}[/tex]
Where, r is the radius and h is the height of the cone.
Putting [tex]r=8,h=15[/tex] in the above formula, we get
[tex]A_1=\pi (8)\sqrt{(15)^2+(8)^2}[/tex]
[tex]A_1=\pi (8)\sqrt{289}[/tex]
[tex]A_1=\pi (8)(17)[/tex]
[tex]A_1=136 \pi [/tex]
The curved surface area of the hemisphere is:
[tex]A_2=2\pi r^2[/tex]
Where, r is the radius.
Putting [tex]r=8[/tex], we get
[tex]A_2=2\pi (8)^2[/tex]
[tex]A_2=2\pi (64)[/tex]
[tex]A_2=128\pi [/tex]
Total surface area of the composite figure is:
[tex]A=A_1+A_2[/tex]
[tex]A=136\pi +128\pi[/tex]
[tex]A=264\pi[/tex]
The total surface area of the composite figure is 264π cm².
Therefore, the correct option is A.
