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Bot GPT 3.5 To find the volume of the entire object (cone + hemisphere), we need to find the volume of the hemisphere and add it to the volume of the cone.
First, we find the radius of the cone using the information given. Since the height of the cone is equal to the diameter of the hemisphere, the radius of the hemisphere is equal to half the height of the cone. So, the radius of the cone is half the diameter, which is half the height of the cone.
Let h be the height of the cone and the diameter of the hemisphere. Therefore, the radius of the cone = h/2.
Given that the volume of the cone is 12 m^3, we can use the formula for the volume of a cone:
V_cone = (1/3) * π * r^2 * h
Substitute the given volume and radius into the formula:
12 = (1/3) * π * (h/2)^2 * h
12 = (1/3) * π * (h^2/4) * h
12 = (Ï€ * h^3) / 12
144 = π * h^3
h^3 = 144 / π
h = (144 / π)^(1/3)
Now that we have found the value of h, we can calculate the volume of the entire object. The volume of a hemisphere is (2/3) * π * r^3, where r = h/2.
V_hemisphere = (2/3) * π * (h/2)^3
Add the volume of the hemisphere to the volume of the cone to get the total volume:
V_total = V_cone + V_hemisphere
V_total = 12 + (2/3) * π * (h/2)^3
Substitute the values of h and r into the formula:
V_total = 12 + (2/3) * π * ((144 / π)^(1/3) / 2)^3
V_total = 12 + (2/3) * π * (72 / π)^(1/3)
V_total = 12 + (48)^(1/3)
V_total is approximately equal to 18.077 m^3.