Hey Bot I need your help figuring out this question.
Can you put it in a short equation?
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.
Let's denote the height and diameter of the ice cream cone as h and d, respectively.
The volume of a cone is given by the formula: V(cone) = (1/3) * π * r^2 * h, where r is the radius of the cone.
Since the diameter of the cone is equal to its height, we have d = h. Therefore, the radius is half the diameter, so r = d/2.
The volume of a sphere is given by the formula: V(sphere) = (4/3) * π * r^3.
We are given that the hemisphere on top of the cone has a volume of 4 in^3. This is half the volume of a complete sphere, so we can write:
4 = (1/2) * [(4/3) * π * (d/2)^3]
Simplifying this equation:
4 = (1/2) * [(4/3) * π * (d^3/8)]
Multiplying through by 2/π:
8/π = (4/3) * (d^3/8)
Multiplying through by 3/4:
6/π = d^3/8
Multiplying through by 8:
48/π = d^3
Taking the cube root of both sides:
d = (48/π)^(1/3)
Now, we can substitute the value of d back into the formula for the volume of the cone:
V(cone) = (1/3) * π * (d/2)^2 * d
Simplifying this equation:
V(cone) = (1/3) * π * (d^2/4) * d
V(cone) = (π/12) * d^3
Now, we can find the total volume of the ice cream in and on top of the cone by adding the volume of the cone and the volume of the hemisphere:
Total volume = V(cone) + V(hemisphere)
Total volume = (π/12) * d^3 + (1/2) * [(4/3) * π * (d/2)^3]
Total volume = (π/12) * d^3 + (2/3) * π * (d/2)^3
Total volume = (π/12) * d^3 + (1/6) * π * d^3
Total volume = (7/12) * π * d^3
Total volume = (7/12) * π * [(48/π)^(1/3)]^3
Total volume = (7/12) * π * (48/π)
Total volume = (7/12) * 48
Total volume = 7 * 4
Total volume = 28 in^3
Therefore, the total volume of ice cream in and on top of the cone is 28 in^3.