Question 842137
the radius and the height of the cone form a triangle which is a right triangle that is in the ratio of 3:4:5.


3 is the height of the cone.
4 is the radius of the cone.
5 is the slant height of the cone.


if we let x equal the common multiple of this ratio, then we get:


3x is the height of the cone.
4x is the radius of the cone.
5x is the slant height of the cone.


if x is equal to 1, then the ratio is 3:4:5.
if x is equal to 2, then the ratio is 6:8:10, etc.


the formula for the total surface area of the cone is equal to pi * r * s + pi * r^2


pi * r * s is the area of the curved surface.


pi * r^2 is the area of the base.


we know the area of the base is equal to 154 cm^2.


we can use that fact to solve for x.


the formula for the area of the base is pi * r^2 = 154


since r is equal to 4x, this means that the formula for the base becomes:


pi * (4x)^2 = 154 which can be simplified to:


pi * 16x^2 = 154


we solve for x in this formula to get:


x = sqrt (154 / (16 * pi) which makes x = 1.750352152


the area of the base is equal to (4x)^2 * pi which is equal to (4 * 1.750352152)^2 * pi which is equal to 154.


the area of the curved surface of the cone is equal to pi * (4x) * (5x) which is equal to pi * (4 * 1.750352152 * 5 * 1.750352152) which is equal to (pi * 4 * 5 * (.750352152)^2) which is equal to 192.5 cm^2.


that's your answer.


the area of the base is equal to 154 cm^2.
the area of the curved surface area is equal to 192.5 cm^2.
the total surface area is equal to 346.5 cm^2.


here a link to where i found the formula for the surface area of a cone.


<a href = "http://math.about.com/od/formulas/ss/surfaceareavol_2.htm" target = "_blank">http://math.about.com/od/formulas/ss/surfaceareavol_2.htm</a>