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 2 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 start by finding the volume of the cone. The formula for the volume of a cone is given by Vcone = 1/3 * π * r^2 * h, where r is the radius and h is the height.
Since the diameter of the cone is equal to its height, let's call the diameter d. This means the radius of the cone is r = d/2.
Now, we need to find the height of the cone. Let's call it h. From the information given, we know that the volume of the hemisphere on top of the cone is 2 in^3. The formula for the volume of a hemisphere is given by Vhemisphere = 2/3 * π * r^3. Let's solve for the radius of the hemisphere.
Vhemisphere = 2 in^3
2/3 * π * r^3 = 2 in^3
(2/3 * π * r^3) / (2/3 * π) = 2 in^3 / (2/3 * π)
r^3 = 1 in^3 / (1/π)
r^3 = π in^3
r = (π in^3)^(1/3)
r = (π in^3)^(1/3) cm
r ≈ 1.464 cm.
Since the radius of the hemisphere is equal to the diameter of the cone, we can say that d = 2r. Substituting in the value of r, we get:
d = 2 * 1.464 cm
d ≈ 2.926 cm.
Because the diameter is equal to the height, we can say that h ≈ 2.926 cm.
Now, let's calculate the volume of the whole cone:
Vcone = 1/3 * π * r^2 * h
Vcone = 1/3 * π * (2.926/2)^2 * 2.926
Vcone ≈ 1/3 * π * 0.927^2 * 2.926
Vcone ≈ 1/3 * π * 0.859 * 2.926
Vcone ≈ 0.865π cm^3.
Finally, let's calculate the total volume of ice cream in and on top of the cone. This is given by the volume of the cone plus the volume of the hemisphere:
Total volume = Vcone + Vhemisphere
Total volume ≈ 0.865π cm^3 + 2 in^3
Total volume ≈ 0.865π cm^3 + 2 cm^3 (since 1 in^3 ≈ 1.64 cm^3)
Total volume ≈ 0.865π cm^3 + 2 cm^3
Total volume ≈ (0.865π + 2) cm^3
Total volume ≈ (2.865π) cm^3.
Therefore, the total volume of ice cream in and on top of the cone is approximately 2.865π cm^3.
Can you make it short
Sure! The volume of a cone is given by Vcone = 1/3 * π * r^2 * h, where r is the radius and h is the height. Since the diameter of the cone is equal to its height, let's call it d. Then we have r = d/2.
The volume of a hemisphere is given by Vhemisphere = 2/3 * π * r^3.
Given that the volume of the hemisphere is 2 in^3, we can calculate its radius as r ≈ 1.464 cm.
Since the radius of the hemisphere is equal to the diameter of the cone, we can say that d ≈ 2.926 cm, which is also the height of the cone.
The volume of the cone is Vcone ≈ 0.865π cm^3.
So, the total volume of ice cream in and on top of the cone is approximately 2.865π cm^3.