A hemisphere is placed on top of an inverted cone. The back of the circumference, the radius, and the perpendicular height of the cone are marked with dashed lines. The radius and the height make a right angle and are perpendicular.
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.
To solve this problem, we need to find the volume of the cone and the volume of the hemisphere, and then add them together.
First, let's find the volume of the cone. We know that the diameter of the cone is equal to its height, so let's call that value "h". The formula for the volume of a cone is V = (1/3)πr^2h, where "r" is the radius. Since the radius is half of the diameter, we can write the formula as V = (1/3)π(h/2)^2h = (1/12)πh^3.
Next, let's find the volume of the hemisphere. The volume of a sphere (which is half of a sphere) is given by the formula V = (4/3)πr^3. Since the diameter of the hemisphere is equal to the height of the cone, we can write the formula as V = (4/3)π(h/2)^3 = (1/6)πh^3.
Now, let's add the volume of the cone and the volume of the hemisphere together. V_total = (1/12)πh^3 + (1/6)πh^3 = (1/12 + 1/6)πh^3 = (1/12 + 2/12)πh^3 = (3/12)πh^3 = (1/4)πh^3.
Since we know that the total volume is 4 in^3, we can set up the equation (1/4)πh^3 = 4 and solve for h:
πh^3 = 16
h^3 = 16/π
h ≈ 2.26015
Now that we know the height of the cone, we can calculate its volume:
V_cone = (1/12)π(2.26015)^3 ≈ 4/3 in^3
And finally, we can calculate the volume of the hemisphere:
V_hemisphere = (1/6)π(2.26015)^3 ≈ 8/3 in^3
Adding the two volumes together, we get the total volume:
V_total = (4/3 + 8/3) in^3 = 12/3 in^3 = 4 in^3
So the total volume of ice cream in and on top of the cone is 4 in^3.
Make it more simple
To simplify the problem, let's assume the diameter and height of the cone are both "d".
The volume of a cone is given by the formula V_cone = (1/3)πr^2h. In this case, the radius of the cone is half of the diameter, so r = d/2. Substituting these values into the formula, we get V_cone = (1/3)π(d/2)^2(d) = (1/12)πd^3.
The volume of a hemisphere (half of a sphere) is given by the formula V_hemisphere = (2/3)πr^3. Since the diameter of the hemisphere is equal to the cone's height (d), the radius is half of the diameter, so r = d/2. Substituting these values into the formula, we get V_hemisphere = (2/3)π(d/2)^3 = (1/6)πd^3.
The total volume of ice cream is the sum of the cone volume and hemisphere volume: V_total = V_cone + V_hemisphere = (1/12)πd^3 + (1/6)πd^3 = (1/12 + 1/6)πd^3 = (1/4)πd^3.
Given that the total volume is 4 in^3, we can set up the equation (1/4)πd^3 = 4 and solve for d^3:
πd^3 = 16
d^3 = 16/π
d ≈ 2.26015
Now we can calculate the volume of the cone:
V_cone = (1/12)π(2.26015)^3 ≈ 4/3 in^3
And the volume of the hemisphere:
V_hemisphere = (1/6)π(2.26015)^3 ≈ 8/3 in^3
The total volume of ice cream in and on top of the cone is the sum of these volumes:
V_total = (4/3 + 8/3) in^3 = 12/3 in^3 = 4 in^3.