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 find the total volume of ice cream in and on top of the cone, we need to find the volume of the cone and add it to the volume of the hemisphere.
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 base and h is the height.
Since the diameter of the ice cream cone is equal to its height, we can write the radius as r = h/2.
Substituting this into the formula for the volume of the cone, we get V_cone = (1/3) * π * (h/2)^2 * h = (1/12) * π * h^3.
The volume of a hemisphere is given by the formula V_hemisphere = (2/3) * π * r^3, where r is the radius.
Since the diameter of the hemisphere is equal to the diameter of the ice cream cone, we can write the radius as r = h/2.
Substituting this into the formula for the volume of the hemisphere, we get V_hemisphere = (2/3) * π * (h/2)^3 = (1/6) * π * h^3.
Adding the volume of the cone and the volume of the hemisphere, we get the total volume of ice cream as:
V_total = (1/12) * π * h^3 + (1/6) * π * h^3 = (1/12 + 1/6) * π * h^3 = (3/12 + 2/12) * π * h^3 = (5/12) * π * h^3.
Since the volume of the hemisphere is given as 4 in^3, we can set the total volume equal to 4 and solve for h:
(5/12) * π * h^3 = 4.
Dividing both sides of the equation by (5/12) * π, we get:
h^3 = (12/5) * 4 / π = 48 / (5 * π).
Taking the cube root of both sides of the equation, we get:
h = ∛(48 / (5 * π)).
So the height of the ice cream cone is ∛(48 / (5 * π)).
To find the total volume of ice cream, we can substitute this value of h into the equation for V_total:
V_total = (5/12) * π * (∛(48 / (5 * π)))^3.
Evaluating this expression will give us the total volume of ice cream in and on top of the cone.
In few sentences
The total volume of ice cream in and on top of the cone can be found by taking the volume of the cone and adding it to the volume of the hemisphere. The cone has a volume of (1/12) * π * h^3 and the hemisphere has a volume of (1/6) * π * h^3. By summing these two volumes, we can find the total volume of ice cream.
To find the total volume of ice cream in and on top of the cone, we need to calculate the volumes of the cone and the hemisphere separately and then add them together.
Let's begin by finding the volume of the cone. We know that the diameter of the ice cream cone is equal to its height. Let's call the height and diameter of the cone "h" and "d" respectively.
We know the formula for the volume of a cone is:
V_cone = 1/3 * π * r^2 * h,
where r is the radius of the base of the cone.
Since the diameter of the cone is equal to its height, we can write:
d = h.
The radius of the base of the cone is half of the diameter, so we have:
r = d/2 = h/2.
Substituting r into the formula for the volume of the cone, we get:
V_cone = 1/3 * π * (h/2)^2 * h
= 1/3 * π * (h^2/4) * h
= π * h^3 / 12.
So, the volume of the cone is π * h^3 / 12.
Now, let's find the volume of the hemisphere. The formula for the volume of a hemisphere is:
V_hemisphere = 2/3 * π * r^3.
Since the diameter of the hemisphere is equal to the height of the cone, we have:
r = d/2 = h/2.
Substituting r into the formula for the volume of the hemisphere, we get:
V_hemisphere = 2/3 * π * (h/2)^3
= π * h^3 / 12.
So, the volume of the hemisphere is π * h^3 / 12.
Finally, to find the total volume of ice cream, we add the volume of the cone and the volume of the hemisphere:
Total volume = V_cone + V_hemisphere
= π * h^3 / 12 + π * h^3 / 12
= 2/12 * π * h^3
= π * h^3 / 6.
Given that the volume of the hemisphere is 4 in^3, we can set up the equation:
π * h^3 / 6 = 4.
To solve for h, we can rearrange the equation:
h^3 = (4 * 6) / π
h^3 = 24 / π
h ≈ ∛(24 / π).
Approximating the value of π as 3.14, we can solve for h:
h ≈ ∛(24 / 3.14)
h ≈ ∛7.644
h ≈ 1.974.
So, the height of the cone is approximately 1.974 inches.
Finally, substituting the value of h into the formula for the total volume, we get:
Total volume ≈ π * (1.974)^3 / 6.
Calculating this value will give us the total volume of ice cream in and on top of the cone.