Aimee packs ice cream into an ice cream cone. She then puts a perfe hemisphere of ice cream on top of the cone that has a volume of 2 in. 3 The diameter of the ice cream cone is equal to its height. W 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 sphe to help solve this problem. Show your work and explain your reasoning
Let's start by finding the formula for the volume of a cone and a hemisphere.
The volume of a cone is given by the formula:
V_cone = (1/3)πr²h
The volume of a hemisphere is given by the formula:
V_hemisphere = (2/3)πr³
Given that the diameter of the ice cream cone is equal to its height, let's call the height and radius of the cone "h" and "r" respectively.
Since the diameter is equal to the height, we have h = 2r.
Now, let's calculate the volume of the cone. We know that Aimee packs ice cream into the cone, so its volume is not empty.
Let's assume the volume of the ice cream in the cone is V_icecream.
Therefore, the volume of the cone with the ice cream is:
V_cone_icecream = V_cone + V_icecream
V_cone_icecream = (1/3)πr²h + V_icecream
Now, let's calculate the volume of the hemisphere. We know that Aimee puts a perfect hemisphere on top of the cone, so the ice cream fills the hemisphere perfectly.
The volume of the hemisphere is given as 2 in³, which we can express as the volume formula:
2 in³ = (2/3)πr³
Now, we have two equations:
V_cone_icecream = (1/3)πr²h + V_icecream
2 in³ = (2/3)πr³
Since h = 2r, we can substitute h in the first equation:
V_cone_icecream = (1/3)πr²(2r) + V_icecream
Simplifying the equation:
V_cone_icecream = (2/3)πr³ + V_icecream
Now, we can equate the equations:
(2/3)πr³ + V_icecream = 2 in³
Subtracting V_icecream from both sides:
(2/3)πr³ = 2 in³ - V_icecream
Simplifying the equation:
(2/3)πr³ = 2 in³ - V_icecream
Multiplying both sides by (3/2)π:
r³ = (3/2)(2 in³ - V_icecream)
r³ = 3 in³ - (3/2)V_icecream
Taking the cube root of both sides:
r = ∛(3 in³ - (3/2)V_icecream)
Now that we have the value of r, we can substitute it back into the equation for V_cone_icecream:
V_cone_icecream = (1/3)πr²h + V_icecream
Since h = 2r:
V_cone_icecream = (1/3)πr²(2r) + V_icecream
Simplifying the equation:
V_cone_icecream = (2/3)πr³ + V_icecream
Substituting r = ∛(3 in³ - (3/2)V_icecream):
V_cone_icecream = (2/3)π(∛(3 in³ - (3/2)V_icecream))³ + V_icecream
Now, you can simplify the equation and solve for V_icecream.
Can you make it shorter
Certainly! We are given that the diameter of the ice cream cone is equal to its height. Let's call the height and radius of the cone "h" and "r" respectively.
The volume of a cone is given by V_cone = (1/3)πr²h.
The volume of a hemisphere is given by V_hemisphere = (2/3)πr³.
Since the cone is filled with ice cream, let's assume the volume of ice cream in the cone is V_icecream. Therefore, the volume of the cone with the ice cream is V_cone_icecream = (1/3)πr²h + V_icecream.
The volume of the hemisphere is 2 in³, which we can express as (2/3)πr³.
We can equate the two equations:
(1/3)πr²h + V_icecream = (2/3)πr³
Since h = 2r, we can substitute h in the equation to get:
(1/3)πr²(2r) + V_icecream = (2/3)πr³
Simplifying further:
(2/3)πr³ + V_icecream = (2/3)πr³
Simplifying again:
V_icecream = 0
This means that the volume of the ice cream in the cone is 0. Therefore, the total volume of ice cream in and on top of the cone is only the volume of the hemisphere, which is 2 in³.
Just a little shorter
Sure! Given that the diameter of the ice cream cone is equal to its height, the volume of ice cream in and on top of the cone can be expressed as the volume of the hemisphere, which is 2 in³.
A little more detail
Certainly!
Given that the diameter of the ice cream cone is equal to its height, let's call the height and radius of the cone "h" and "r" respectively.
The volume of a cone is given by the formula V_cone = (1/3)πr²h, where "r" is the radius and "h" is the height.
The volume of a hemisphere is given by the formula V_hemisphere = (2/3)πr³, where "r" is the radius.
Since the diameter of the cone is equal to its height, we have h = 2r.
Therefore, the volume of the ice cream in and on top of the cone can be expressed as the volume of the hemisphere on top of the cone. In other words, it is the volume of a perfect hemisphere with a radius of "r".
Substituting the value of "r" from the height of the ice cream cone, we have:
V_hemisphere = (2/3)π(2r)³
V_hemisphere = (2/3)π(8r³)
V_hemisphere = (16/3)πr³
So, the total volume of ice cream in and on top of the cone is (16/3)πr³.
Note: If you have the value of the radius "r", you can substitute it in the formula to calculate the exact volume in cubic inches.