Question

Find the lateral area, surface area, and volume?

Answer 1

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`