An engineer wants to know the volume of the following object composed of a
hemisphere and a cone. She knows the height of the cone is equal to the
diameter of the hemisphere, and the volume of the cone is 12 m³. What is the
volume of the whole object? (1 point)
To find the volume of the whole object, we need to sum the volumes of the hemisphere and the cone.
Since the volume of the cone is given as 12 m³, we can use the formula for the volume of a cone:
Vcone = 1/3 * π * r² * h
Where r is the radius of the cone and h is the height of the cone.
Given that the height of the cone is equal to the diameter of the hemisphere, we know that the radius of the cone is half the diameter of the hemisphere.
Let's denote the radius of the cone as r and the height of the cone as d (diameter of the hemisphere).
So, the volume of the cone can be written as:
12 = 1/3 * π * r² * d
Now, let's find a relationship between d and r:
We know that the height of the cone (d) is twice the radius of the cone (r), since the height of the cone is equal to the diameter of the hemisphere.
d = 2r
Now we can substitute this relation into the volume formula for the cone:
12 = 1/3 * π * r² * 2r
Rearranging this equation, we get:
r³ = 6 / π
Now, the volume of the hemisphere can be calculated using the formula:
Vhemisphere = 2/3 * π * r³
Substitute the value of r³ from above:
Vhemisphere = 2/3 * π * 6 / π
Vhemisphere = 4 * 6 = 24 m³
Therefore, the volume of the whole object (hemisphere + cone) is:
Vwhole = Vhemisphere + Vcone
Vwhole = 24 + 12
Vwhole = 36 m³
Therefore, the volume of the whole object is 36 m³.