Find the lateral area, surface area, and volume?

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1 Answer
Jun 18, 2018

Lateral Area: L.A. = 6pic
Total surface Area: A_T = 6pic + 2pic^2
Total volume: V_T = pic^2(sqrt(36-c^2))/3 + 2/3pic^3

Explanation:

The description of the problem is not very clear, but I'll give it a shot, with some assumptions. I will assume that the shape is a right circular cone, inverted, and with a half sphere on top. I will also assume that the "c" refers to the radius of the sphere, which is also the radius of the base of the cone, and the "6" refers to the slanted height of the cone.

The lateral area of a right circular cone is given by:
L.A. = pi*r*s
where r is the radius, and s is the slanted height
So, here: L.A. = pi*c*6 = 6pic

For the total surface area, we need to add the surface area of the half-sphere. The surface area of a whole sphere is given by: A_s = 4pir^2, which means that the exposed surface area of our half sphere is A_h = 1/2*4pic^2=2pic^2
If we add this to the lateral area, we get the total surface area:
A_T = 6pic + 2pic^2

For the volume, let's first figure out the volume of the cone, which is given by: V_c = pir^2h/3

From Pythagoras: h^2+c^2=6^2
so, h=sqrt(36-c^2)

So, the volume of the cone: V_c = pic^2(sqrt(36-c^2))/3

The volume of a sphere is V_s=4/3pir^3
so, the volume of our half-sphere is: V_h = 1/2*4/3pic^3=2/3pic^3

Adding the 2 volumes, we get the total volume of the shape:
V_T = pic^2(sqrt(36-c^2))/3 + 2/3pic^3