Answer :
The radius of the cone is 16.33, the height of the cone is 11.55, and the maximum volume of the cone is 1,026.67[tex]\pi[/tex].
How to find volume of the cone?
Given
metal radius = R = 20
cone base radius = r
cone height = h
First, we can use Pythagoras theorem
R^2 = r^2 + h^2
r^2 = R^2 - h^2
r^2 = 400 - h^2
Then, we use volume of cone formula
V = 1/3[tex]\pi[/tex] * r^2 * h
V = 1/3[tex]\pi[/tex] * (400 - h^2) * h
V = 1/3[tex]\pi[/tex] * 400h - h^3
To get maximum volume of cone, V'(h) must be 0. So,
1/3[tex]\pi[/tex] * 400 - 3h^2 = 0
400 - 3h^2 = 0
3h^2 = 400
h^2 = 400/3
h = [tex]\frac{20}{\sqrt{3}}[/tex] or 11.55
Next, we find the r with substitution method. So,
r^2 = 400 - ([tex]\frac{20}{\sqrt{3}}[/tex] )^2
r^2 = 400 - [tex]\frac{400}{3}[/tex]
r^2 = [tex]\frac{800}{3}[/tex]
r = [tex]\frac{20\sqrt{2}}{\sqrt{3}}[/tex] or 16.33
Now, we can get maximum volume of cone.
V = 1/3[tex]\pi[/tex] * 16.33^2 * 11.55
V = 1,026.67[tex]\pi[/tex]
Thus, the radius of the cone is 16.33, the height of the cone is 11.55, and the maximum volume of the cone is 1,026.67[tex]\pi[/tex].
Learn more about volume here:
brainly.com/question/13338592
#SPJ4
