An engineer wants to know the volume of the following object composed of a hemisphere in a cone he knows the height of the cone is equal to the diameter of the hemisphere and the volume of the cone is 12 M cubed what is the volume of the whole object
To find the volume of the whole object, we need to calculate the volumes of the hemisphere and the cone separately and then add them together.
The volume of a cone is given by the formula Vcone = (1/3)πr^2h, where r is the radius and h is the height. We are given that the volume of the cone is 12 m^3.
The volume of a hemisphere is given by the formula Vhemisphere = (2/3)πr^3, where r is the radius. In this case, the height of the cone is equal to the diameter of the hemisphere, which means the height of the cone is twice the radius of the hemisphere. Thus, the radius of the hemisphere is h/2.
Let's start by finding the radius of the cone. We can rearrange the formula for the volume of a cone to solve for the radius:
12 = (1/3)πr^2h
36 = πr^2h
We know that h = 2r (since the height of the cone is equal to the diameter of the hemisphere), so we can substitute this into the equation:
36 = πr^2(2r)
36 = 2πr^3
18 = πr^3
r^3 = 18/π
r ≈ 2.062
Now we can find the height of the cone:
h = 2r
h ≈ 4.125
The volume of the cone is given as 12 m^3, so we can now calculate the value of π:
Vcone = (1/3)πr^2h
12 = (1/3)π(2.062)^2(4.125)
36 = π(8.51)(4.125)
36 = π(35.12325)
36/π ≈ 11.459
Now we can find the volume of the whole object by adding the volume of the cone and the hemisphere together:
Vobject = Vcone + Vhemisphere
Vobject = 11.459 + (2/3)π(2.062)^3
Vobject ≈ 11.459 + (2/3)π(8.51)
Vobject ≈ 11.459 + (2/3)π(72.24075)
Vobject ≈ 11.459 + π(48.1605)
Vobject ≈ 11.459 + 150.657
Vobject ≈ 162.116
Therefore, the volume of the whole object is approximately 162.116 m^3.