Respuesta :
Answer:
The surface area of a cone of radius r and height h not equal​ to one-third the surface area of a cylinder with the same radius and​ height.
Relationship is [tex]S_c=(\frac{\sqrt{(r+h)}}{2h})S_C[/tex]
Step-by-step explanation:
Given : The volume of a cone of radius r and height h is​ one-third the volume of a cylinder with the same radius and height.
To find : Does the surface area of a cone of radius r and height h equal​ one-third the surface area of a cylinder with the same radius and​ height?
If​ not, find the correct relationship. Exclude the bases of the cone and cylinder.
Solution :
Radius of cone and cylinder is 'r'.
Height of cone and cylinder is 'h'.
The volume of cone is [tex]V_c=\frac{1}{3}\pi r^2 h[/tex]
The volume of cylinder is [tex]V_C=\pi r^2 h[/tex]
[tex]\frac{V_c}{V_C}=\frac{\frac{1}{3}\pi r^2 h}{\pi r^2 h}[/tex]
[tex]V_c=\frac{1}{3}V_C[/tex]
i.e. volume of cone is one-third of the volume of cylinder.
Now,
Surface area of the cone is [tex]S_c=\pi r\sqrt{(r+h)}[/tex]
Surface area of the cylinder is [tex]S_C=2\pi rh[/tex]
Dividing both the equations,
[tex]\frac{S_c}{S_C}=\frac{\pi r\sqrt{(r+h)}}{2\pi rh}[/tex]
[tex]\frac{S_c}{S_C}=\frac{\sqrt{(r+h)}}{2h}[/tex]
[tex]S_c=(\frac{\sqrt{(r+h)}}{2h})S_C[/tex]
Which clearly means [tex]S_c\neq \frac{1}{3}S_C[/tex]
i.e. The surface area of a cone of radius r and height h not equal​ to one-third the surface area of a cylinder with the same radius and​ height.
The relationship between them is
[tex]S_c=(\frac{\sqrt{(r+h)}}{2h})S_C[/tex]
