The ratio of the volume to the lateral area of a right circular cone is 2:1. If the altitude is 15cm, what is the ratio of the slant height to the radius?
To find the ratio of the slant height to the radius, we can start by finding the formulas for the volume, the lateral area, and the slant height of a right circular cone.
The volume of a cone is given by the formula:
V = (1/3) * π * r^2 * h
where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the radius, and h is the height.
The lateral area of a cone is given by the formula:
L = π * r * s
where L is the lateral area, r is the radius, and s is the slant height.
Given that the ratio of the volume to the lateral area is 2:1, we can set up the equation:
V / L = 2 / 1
Substituting the volume and lateral area formulas, we get:
[(1/3) * π * r^2 * h] / (π * r * s) = 2 / 1
Simplifying the equation:
(1/3) * r * h / s = 2
Now we can substitute the given values:
(1/3) * r * 15 / s = 2
Simplifying further:
5 * r / s = 2
Finally, rearranging the equation to isolate the ratio of the slant height to the radius, we have:
s / r = 5 / 2
Therefore, the ratio of the slant height to the radius is 5:2.