A right circular cone has the property that its volume is equal to 1.5 times its lateral surface area and its
radius is half of its slant height. Compute the radius of this cone.
s = 2r
but
r^2 + h^2 = s^2 = 4r^2
so h^2 = 3r^2
h = √3 r
Now, we can solve
1/3 πr^2 h = 3/2 πrs
so, what do you get?
Let's assume the radius of the cone is "r" units and the slant height is "s" units.
The formula for the volume of a cone is given by: V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the height.
The formula for the lateral surface area of a cone is given by: L = πrs, where L is the lateral surface area, r is the radius, and s is the slant height.
Given that the volume is equal to 1.5 times the lateral surface area, we can write the equation:
(1/3)πr^2h = 1.5(πrs)
Since the radius is half of the slant height, we can write:
r = s/2
Substituting this back into the equation, we get:
(1/3)π(s/2)^2h = 1.5(π(s/2)s)
Simplifying further:
(1/3)(s^2/4)h = 1.5(s^2/2)
Multiplying both sides by 12 to clear the fractions:
3(s^2/4)h = 9(s^2/2)
Simplifying:
(3/4)sh = 9/2 s^2
Dividing both sides by s:
(3/4)h = (9/2)s
Since r = s/2, we can write:
(3/4)h = (9/2)(2r)
Simplifying:
(3/4)h = 9r
Dividing both sides by 9:
(1/4)h = r
Therefore, the radius of the cone is (1/4) times the height.
To solve this problem, we need to use the given information about the volume, lateral surface area, and the relationship between the radius and the slant height of the cone.
Let's start by expressing the volume and lateral surface area of the cone in terms of its radius and height.
The volume of a cone can be expressed as V = (1/3) * π * r^2 * h, where r is the radius and h is the height.
The lateral surface area of a cone can be expressed as A = π * r * L, where L is the slant height.
Given that the volume of the cone is equal to 1.5 times its lateral surface area, we can write the equation:
(1/3) * π * r^2 * h = 1.5 * π * r * L
Since we also know that the radius is half of the slant height, we can write:
r = L/2
Substituting this into the previous equation, we get:
(1/3) * π * (L/2)^2 * h = 1.5 * π * (L/2) * L
Simplifying and canceling out common terms, we are left with:
(L^2 * h) / 12 = (3L^2) / 4
Cross-multiplying, we get:
4 * L^2 * h = 36 * L^2
Dividing both sides by L^2, we get:
4h = 36
Dividing both sides by 4, we find:
h = 9
Now, we can substitute the value of h back into our expression for the radius:
r = L/2
Since r = L/2 and h = 9, we have:
r = L/2 = 9/2 = 4.5
Therefore, the radius of the cone is 4.5 units.