suppose a sphere, cylinder, and cone all share the same radius and the cylinder and cone also share the same height which is twice the radius. which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone.
A. sphere - cone + cylinder
B. 3 cones = 1 cylinder
C. sphere = cylinder + cone
D. 1 cylinder = 2 cones + 1 sphere
The correct answer is D. 1 cylinder = 2 cones + 1 sphere.
This can be derived from the volume formulas for each shape:
- The volume of a sphere is given by V_sphere = (4/3)πr^3.
- The volume of a cylinder is given by V_cylinder = πr^2h, where h is the height of the cylinder.
- The volume of a cone is given by V_cone = (1/3)πr^2h.
Given that the radius is the same for all three shapes and the height of the cylinder and cone is twice the radius, we have h = 2r.
Plugging this value into the volume formulas, we get:
V_sphere = (4/3)πr^3
V_cylinder = πr^2(2r) = 2πr^3
V_cone = (1/3)πr^2(2r) = (2/3)πr^3
Now we can see that 1 cylinder is equal to 2 cones plus 1 sphere:
V_cylinder = V_cone + V_cone + V_sphere
2πr^3 = (2/3)πr^3 + (2/3)πr^3 + (4/3)πr^3
2πr^3 = (4/3)πr^3 + (4/3)πr^3
2πr^3 = (8/3)πr^3
Thus, 1 cylinder is equal to 2 cones plus 1 sphere, which leads us to option D.
the volume of a sphere is 48m^3. what is the volume of a cylinder if its radius is the same as the spheres and its height is equal to the spheres diameter.
To find the volume of the cylinder, you first need to determine its height. Given that the height of the cylinder is equal to the diameter of the sphere, we can calculate the diameter using the formula:
Diameter = 2 * Radius
Since the radius is the same as the sphere, the diameter of the sphere is also equal to the radius. Therefore, the diameter is equal to the radius.
Given that the volume of the sphere is 48 m^3, we can write the formula for the volume of the sphere:
V_sphere = (4/3)πr^3
Substituting the volume and rearranging the formula, we have:
48 = (4/3)πr^3
To solve for r, we can divide both sides of the equation by (4/3)π:
r^3 = 48 * (3/4) / π
r^3 = 36 / π
Now, let's find the volume of the cylinder. The volume of a cylinder is given by the formula:
V_cylinder = πr^2h
Since the height of the cylinder is equal to the diameter of the sphere, which is equal to the radius of the sphere, the height of the cylinder is r as well.
Substituting the value of r into the formula, we have:
V_cylinder = πr^2 * r = πr^3
Using the value we found for r^3 earlier, we can calculate the volume of the cylinder:
V_cylinder = π * (36 / π) = 36 m^3
Therefore, the volume of the cylinder is 36 m^3.
aimee packs ice cream into an ice cream cone. she then puts a perfect hemisphere of ice cream on top of the cone that has a volume of 14in^3. the diameter of the ice cream cone is equal to its height. what is the total volume of ice cream in and on top of the cone? use the relationship between the formulas for the volumes of cones and spheres to help solve this problem. show your work and explain your reasoning
To find the total volume of ice cream in and on top of the cone, we can break it down into two parts: the volume of the ice cream in the cone and the volume of the hemisphere on top.
1. Volume of ice cream in the cone:
Given that the diameter of the cone is equal to its height, let's assume that the radius of the cone is r. The volume of a cone is given by the formula:
V_cone = (1/3) * π * r^2 * h
Since the height and diameter are equal, we have h = 2r. Substituting this into the volume formula, we get:
V_cone = (1/3) * π * r^2 * 2r
= (2/3) * π * r^3
2. Volume of the hemisphere on top:
The volume of a hemisphere is given by the formula:
V_hemisphere = (2/3) * π * r^3
Given that the volume of the hemisphere is 14 in^3, we can write:
(2/3) * π * r^3 = 14
To find the value of r, let's solve this equation:
π * r^3 = 14 / (2/3)
r^3 = (14 / (2/3)) / π
r^3 = 21π / 4
r ≈ 1.683 in (approximately)
Now, let's calculate the total volume:
V_total = V_cone + V_hemisphere
= (2/3) * π * r^3 + (2/3) * π * r^3
= (2/3) * π * (r^3 + r^3)
= (2/3) * π * (2r^3)
= (2/3) * π * (2 * 1.683^3)
≈ 11.36 in^3 (approximately)
Therefore, the total volume of ice cream in and on top of the cone is approximately 11.36 in^3.