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.
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius 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.
We are given that the diameter of the ice cream cone is equal to its height. Let's call this value d.
Since the diameter is equal to 2 times the radius, we can say that the radius of the cone and the height of the cone are both d/2.
Using the formula for the volume of a cone, we can calculate its volume:
V_cone = (1/3)π(d/2)^2(d/2)
= (1/3)π(d^2/4)(d/2)
= (1/3)π(d^3/8)
Given that the volume of the hemisphere of ice cream on top of the cone is 4 inches cubed, we can set up the following equation:
(4/3)π(d/2)^3 = 4
Simplifying this equation, we get:
(1/3)π(d^3/8) = 4
π(d^3/8) = 12
d^3/8 = 12/π
d^3 = 96/π
d = (96/π)^(1/3)
Now we can substitute this value of d back into the equation for the volume of the cone:
V_cone = (1/3)π((96/π)^(1/3))^3/8
= (1/3)π(96/π)^(3/8)
= (1/3)(96/π)^(3/8)π
Therefore, the total volume of ice cream in and on top of the cone is:
V_total = V_cone + V_hemisphere
= (1/3)(96/π)^(3/8)π + 4
= (1/3)(96/π)^(3/8)π + 48/12
= (1/3)(96/π)^(3/8)π + 4
= (1/3)(96/π)^(3/8)π + 12/3
= (1/3)(96/π)^(4/8)π + 12/3
= (1/3)(96/π)^(1/2)π + 12/3
= (1/3)(96/π)^(1/2)π + 12/3
= (1/3)(9.68)^(1/2)π + 4
≈ 10.18 + 4
≈ 14.18 inches cubed
I do not agree with your answer. the formula for the volume of ice cream in the cone is the same as the formula for the volume of the hemisphere of ice cream. Because the volume of the cone of ice cream is equal to the volume of the hemisphere of ice cream.
Given the volume of the hemisphere is 4 in³, the total volume of the ice cream is 8 in³.