Find the lateral area, surface area, and volume?

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1 Answer
Jun 18, 2018

Lateral Area: L.A. = 6picL.A.=6πc
Total surface Area: A_T = 6pic + 2pic^2AT=6πc+2πc2
Total volume: V_T = pic^2(sqrt(36-c^2))/3 + 2/3pic^3VT=πc236c23+23πc3

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"c refers to the radius of the sphere, which is also the radius of the base of the cone, and the "6"6 refers to the slanted height of the cone.

The lateral area of a right circular cone is given by:
L.A. = pi*r*sL.A.=πrs
where rr is the radius, and ss is the slanted height
So, here: L.A. = pi*c*6 = 6picL.A.=πc6=6πc

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^2As=4πr2, which means that the exposed surface area of our half sphere is A_h = 1/2*4pic^2=2pic^2Ah=124πc2=2πc2
If we add this to the lateral area, we get the total surface area:
A_T = 6pic + 2pic^2AT=6πc+2πc2

For the volume, let's first figure out the volume of the cone, which is given by: V_c = pir^2h/3Vc=πr2h3

From Pythagoras: h^2+c^2=6^2h2+c2=62
so, h=sqrt(36-c^2)h=36c2

So, the volume of the cone: V_c = pic^2(sqrt(36-c^2))/3Vc=πc236c23

The volume of a sphere is V_s=4/3pir^3Vs=43πr3
so, the volume of our half-sphere is: V_h = 1/2*4/3pic^3=2/3pic^3Vh=1243πc3=23πc3

Adding the 2 volumes, we get the total volume of the shape:
V_T = pic^2(sqrt(36-c^2))/3 + 2/3pic^3VT=πc236c23+23πc3