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 4in³ 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
Short answer
To solve this problem, let's first assume that the height and diameter of the ice-cream cone are both "h" inches.
The formula for the volume of a cone is given by:
V_cone = (1/3) * π * r^2 * h
where r is the radius of the base of the cone.
Since the diameter is equal to the height, the radius is equal to h/2. So the volume of the cone becomes:
V_cone = (1/3) * π * (h/2)^2 * h
V_cone = (1/3) * π * (h^3/4)
The volume of a hemisphere is given by:
V_hemisphere = (2/3) * π * r^3
Since the diameter of the hemisphere is equal to the height of the cone, the radius is also equal to h/2. So, the volume of the hemisphere becomes:
V_hemisphere = (2/3) * π * (h/2)^3
V_hemisphere = (2/3) * π * (h^3/8)
We are given that the volume of the hemisphere is 4in³. Setting up an equation:
4 = (2/3) * π * (h^3/8)
To solve for h, we can simplify the equation:
4 = (h^3 * π)/12
48 = h^3 * π
h^3 = 48/π
h ≈ 3.669
Now that we know the height of the cone, we can substitute it into the formula for the volume of the cone to find the total volume of the ice cream:
V_cone = (1/3) * π * (3.669/2)^2 * 3.669
V_cone ≈ 11.768 in³
The total volume of ice-cream in and on top of the cone is the sum of the volume of the cone and the hemisphere:
Total volume ≈ 11.768 + 4
Total volume ≈ 15.768 in³
Therefore, the total volume of ice-cream is approximately 15.768 cubic inches.
shorter
To solve this problem, we can use the formula for the volume of a cone, which is (1/3) * π * r^2 * h, where r is the radius of the cone's base and h is its height. Since the diameter of the ice-cream cone is equal to its height, the radius is h/2.
The volume of the cone is given by (1/3) * π * (h/2)^2 * h, which simplifies to (1/12) * π * h^3.
The volume of the hemisphere is (2/3) * π * r^3, where r is the radius of the hemisphere. Since the diameter of the hemisphere is equal to the height of the cone, the radius is also h/2.
The volume of the hemisphere is therefore (2/3) * π * (h/2)^3, which simplifies to (1/12) * π * h^3.
If the volume of the hemisphere is 4in³, then:
(1/12) * π * h^3 = 4
π * h^3 = 48
h^3 ≈ 15.26
h ≈ 2.63
Substituting this value of h into the formula for the volume of the cone, we get:
(1/12) * π * (2.63)^3 ≈ 2.86 in³
The total volume of ice cream is the sum of the volume of the cone and the hemisphere:
2.86 + 4 = 6.86 in³
Therefore, the total volume of ice cream in and on top of the cone is approximately 6.86 cubic inches.