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 4 inches cubed. 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 reasoning, but keep it short and simple
We determine the height of the cone and find it to be 1 inch. Using the volume formulas for cones and spheres, we calculate the volume of the cone to be π/12 inches cubed. The volume of the hemisphere on top is given as 4 inches cubed. Adding these volumes together, we find the total volume of ice cream in and on top of the cone to be (π + 48)/12 inches cubed.
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere.
Given that the diameter of the cone is equal to its height, we can say that the radius of the cone (r_cone) is half the height of the cone (h_cone/2).
Let's assume the height of the cone is h_cone. Then, the radius of the cone is r_cone = h_cone/2.
We are also given that the volume of the hemisphere on top of the cone is 4 inches cubed. The volume of the hemisphere is half the volume of a sphere with the same radius. Therefore, the volume of the sphere is 2 times the volume of the hemisphere, which is 2 * 4 inches cubed = 8 inches cubed.
Since the radius of the sphere (r_sphere) is the same as the radius of the cone (r_cone), we can equate the volumes of the cone and the sphere:
(1/3)π(r_cone)^2h_cone = (4/3)π(r_cone)^3
Canceling out the common terms π(r_cone)^2, we have:
(1/3)h_cone = (4/3)(r_cone)^2
Multiplying both sides by 3, we have:
h_cone = 4(r_cone)^2
Substituting r_cone = h_cone/2, we have:
h_cone = 4(h_cone/2)^2
Simplifying, we have:
h_cone = 4(h_cone)^2/4
Cancelling out the common factor of 4, we have:
h_cone = h_cone^2
Taking the square root of both sides, we have:
√h_cone = h_cone
Squaring both sides, we have:
h_cone = h_cone^2
This equation holds true when h_cone = 1, so the height of the cone is 1 inch.
Now, substituting the value of h_cone into the volume formula for a cone, we have:
V_cone = (1/3)π(r_cone)^2h_cone
= (1/3)π(h_cone/2)^2h_cone
= (1/3)π(1/2)^2 * 1
= (1/3)π(1/4)
= π/12
Therefore, the volume of the ice cream in the cone is π/12 inches cubed.
The volume of the ice cream on top of the cone is the volume of the hemisphere, which is given as 4 inches cubed.
So, the total volume of ice cream in and on top of the cone is π/12 + 4 = (π + 48)/12 inches cubed.
make it shorter
The total volume of ice cream in and on top of the cone is (π + 48)/12 inches cubed.
explain your reasoning and answer, but keep it short
Using the relationship between the formulas for the volumes of cones and spheres, we determine that the total volume of ice cream in and on top of the cone is (π + 48)/12 inches cubed.
longer
To find the total volume of ice cream in and on top of the cone, we first need to determine the height of the cone. Given that the diameter of the cone is equal to its height, we can equate the volumes of the cone and the hemisphere on top. By simplifying the equation, we find that the height of the cone is 1 inch.
With the height known, we can calculate the volume of the cone using the formula V = (1/3)πr^2h, where r is the radius and h is the height. The volume of the cone is π/12 inches cubed.
The volume of the hemisphere on top of the cone is already given as 4 inches cubed.
By adding the volumes of the cone and the hemisphere, we find that the total volume of ice cream in and on top of the cone is (π + 48)/12 inches cubed.