Suppose a cone and a cylinder have the same radius and that the slant height l of the cone is the same as the height h of the cylinder. Find the ratio of the cone's surface area to the cylinder's surface area.
Area of cone: S = πrl + πr^2
Area of cylinder: S = 2πrh + 2πr^2
just divide the two, as you would do with any ratio:
(πrl + πr^2) / (2πrh + 2πr^2)
πr(l+r) / 2πr(h+r)
= (l+r) / 2(h+r)
To find the ratio of the cone's surface area to the cylinder's surface area, we need to compare the two expressions for surface area.
Given that the cone and cylinder have the same radius (r), we can substitute the height (h) of the cylinder with the slant height (l) of the cone.
For the cone:
Surface Area = πrl + πr^2
For the cylinder:
Surface Area = 2πrh + 2πr^2
Since l is equal to h, we can substitute l for h in the cylinder's surface area formula:
Surface Area = 2πrl + 2πr^2
Now we have the surface area of both the cone and the cylinder in terms of π, r, and l.
To find the ratio, we divide the surface area of the cone by the surface area of the cylinder:
Ratio = (Surface Area of Cone) / (Surface Area of Cylinder)
= (πrl + πr^2) / (2πrl + 2πr^2)
We can simplify this ratio by factoring out πr:
Ratio = (πr(l + r)) / (2πr(l + r))
= (πr(l + r)) / (2πr(l + r))
= 1/2
Therefore, the ratio of the cone's surface area to the cylinder's surface area is 1/2.