A solid metal cone has radius 10cm and height 36cm.
(i) Calculate the volume of this cone.
[The volume, V, of a cone with radius r and height h is V= 1/3 * pi * r ^ 2 * h .]
(ii) The cone is cut, parallel to its base, to give a smaller cone.
The volume of the smaller cone is half the volume of the original cone. The smaller cone is melted down to make two different spheres. The ratio of the radii of these two spheres is 1/2
Calculate the radius of the smaller sphere.
[The volume, V, of a sphere with radius r is V= 4/3 * pi * r ^ 3 .]
(i) The volume of the cone can be calculated using the given formula:
V = (1/3) * π * r^2 * h
Substituting the given values, we have:
V = (1/3) * π * (10 cm)^2 * 36 cm
Calculating this expression, we have:
V ≈ 3769.91 cm^3
Therefore, the volume of the cone is approximately 3769.91 cm^3.
(ii) The volume of the smaller cone is half the volume of the original cone, so we have:
V(smaller cone) = (1/2) * V(original cone)
Given that the radius of the smaller cone is r, and using the formula for the volume of a cone, we can write:
(1/3) * π * r^2 * h = (1/2) * (1/3) * π * (10 cm)^2 * 36 cm
Simplifying this equation, we have:
r^2 * h = (1/2) * (10 cm)^2 * 36 cm
r^2 * h = 900 cm^3
The ratio of the radii of the two spheres is 1/2. Let's denote the radius of the smaller sphere as r2. Since the ratio of the radii is 1/2, we have:
r2 = (1/2) * r
Substituting this into the equation for the volume of a sphere, we get:
(4/3) * π * r2^3 = (4/3) * π * ((1/2) * r)^3
Simplifying this expression, we have:
(4/3) * π * r2^3 = (4/3) * π * (1/8) * r^3
Canceling out common factors, we get:
r2^3 = (1/8) * r^3
Taking the cube root of both sides, we have:
r2 = (1/2) * r
Comparing this equation with the equation r2 = (1/2) * r obtained earlier, we can see that the radius of the smaller sphere is the same as the radius of the smaller cone.
Therefore, the radius of the smaller sphere is also equal to half the radius of the original cone.
The radius of the smaller sphere is 10 cm / 2 = 5 cm.