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 in.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.
(4 points)
To solve this problem, we can use the relationship between the formulas for the volumes of cones and spheres.
The formula for the volume of a cone is V = 1/3 * π * r^2 * h, where V is the volume, π is a constant (approximately 3.14159), r is the radius of the base of the cone, and h is the height of the cone.
The formula for the volume of a sphere is V = 4/3 * π * r^3, where V is the volume, π is a constant (approximately 3.14159), and r is the radius of the sphere.
In this problem, we are given that the diameter of the cone is equal to its height. Let's call this common value x.
The radius of the base of the cone is half the diameter, so it is x/2.
Using the formula for the volume of a cone, we have:
V_cone = 1/3 * π * (x/2)^2 * x
Simplifying this expression, we get:
V_cone = 1/3 * π * (x/2)^2 * x
= 1/3 * π * (x^2/4) * x
= π/12 * x^3
Now, the volume of the hemisphere on top of the cone is given as 4 in^3.
Using the formula for the volume of a sphere, we have:
V_sphere = 4/3 * π * r^3
The radius of the hemisphere is the same as the radius of the base of the cone, so it is x/2.
Substituting this value into the formula, we have:
4 = 4/3 * π * (x/2)^3
4 = 4/3 * π * (x^3/8)
4 = π/6 * (x^3/2)
x^3 = 8 * (4/π) * 3/2
x^3 = 16 * (3/2)
x^3 = 24
Taking the cube root of both sides, we get:
x = ∛24
x ≈ 2.884
Now that we have the value of x, we can substitute it back into the expression for the volume of the cone to find the total volume of ice cream in and on top of the cone:
V_total = V_cone + V_sphere
= π/12 * x^3 + 4
≈ π/12 * (2.884)^3 + 4
≈ π/12 * 23.093 + 4
≈ 19.082 + 4
≈ 23.082
Therefore, the total volume of ice cream in and on top of the cone is approximately 23.082 cubic inches.