An ice-cream shop offers a small size cone and a large size cone.
The height of the large cone is 2 times the height of the small cone.
The radius of the large cone is 2 times the radius of the small cone.
Which statement is true about the volumes of the two cones?
A. The volume of the large cone is 1/8 times the volume of the small cone.
B. The volume of the large cone is 4 times the volume of the small cone.
C. The volume of the large cone is 6 times the volume of the small cone.
D. The volume of the large cone is 8 times the volume of the small cone.
1/8 times the small
To determine the relationship between the volumes of the small and large cones, we can use the formula for the volume of a cone:
V = (1/3) * π * r^2 * h,
where V is the volume, π is a constant (approximately 3.14159), r is the radius, and h is the height of the cone.
We are given that the height of the large cone is 2 times the height of the small cone, and the radius of the large cone is 2 times the radius of the small cone.
Let's represent the height of the small cone as h_s and the radius as r_s. So, the height of the large cone would be 2 * h_s, and the radius of the large cone would be 2 * r_s.
Now, let's substitute these values into the volume formulas for the small and large cones:
For the small cone:
V_s = (1/3) * π * (r_s)^2 * h_s.
For the large cone:
V_l = (1/3) * π * (2 * r_s)^2 * (2 * h_s).
Simplifying the equation for the volume of the large cone:
V_l = (1/3) * π * (4 * r_s^2) * (2 * h_s).
V_l = (4/3) * π * r_s^2 * (2 * h_s).
V_l = (8/3) * π * r_s^2 * h_s.
Comparing the volumes of the small and large cones, we can see that the volume of the large cone is (8/3) times the volume of the small cone.
Therefore, the correct statement about the volumes of the two cones is option D. The volume of the large cone is 8 times the volume of the small cone.