SOLUTION: A solid cone has a height of 18cm and a radius of 6cm. From the apex, a shape to a height of 12cm is cut off. Find the surface area and volume of the remaining shape.

Algebra ->  Surface-area -> SOLUTION: A solid cone has a height of 18cm and a radius of 6cm. From the apex, a shape to a height of 12cm is cut off. Find the surface area and volume of the remaining shape.       Log On


   

Question 1125741: A solid cone has a height of 18cm and a radius of 6cm. From the apex, a shape to a height of 12cm is cut off. Find the surface area and volume of the remaining shape.
Answer by greenestamps(13200) About Me  (Show Source):
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The shape that is cut off is a cone similar to the original cone. The height of the original cone is 18cm; the height of the cone that is cut off is 12cm. So the scale factor (ratio of similarity) between the smaller and larger cones is 2:3.

By a general principle regarding similar figures, the ratio of surface areas of the two cones will then be 2^2:3^2 = 4:9, and the ratio of volumes will be 2^3:3^3 = 8:27.

The volume of the original cone is

%281%2F3%29%28pi%29%28r%5E2%29%28h%29+=+%281%2F3%29%28pi%29%286%5E2%29%2818%29+=+216%28pi%29

So the volume of the cone that is cut off is

216%28pi%29%2A%288%2F27%29+=+64%28pi%29

Then the volume of the remaining frustum is

216%28pi%29-64%28pi%29+=+152%28pi%29

The surface area of the frustum is the area of the two bases, plus the portion of the lateral surface area of the original cone that is left after the small cone is cut off.

Using the ratio of similarity of the two cones, the radii of the two bases of the frustum are 6cm and 4cm; the area of those bases is

%28pi%29%286%5E2%29%2B%28pi%29%284%5E2%29+=+52%28pi%29

The lateral surface area of the original cone is

%28pi%29%28r%29%28l%29

where l is the slant height. The slant height is found from the Pythagorean Theorem using the radius and height:

l+=+sqrt%286%5E2%2B18%5E2%29+=+sqrt%28360%29+=+6%2Asqrt%2810%29

The lateral surface area of the original cones is then

%28pi%29%286%29%286%2Asqrt%2810%29%29+=+36%28pi%29%2Asqrt%2810%29

The lateral surface area of the small cone that is cut off is 4/9 of that, leaving 5/9 of that as the lateral surface area of the frustum. So the lateral surface area of the frustum is

%285%2F9%29%2A36%28pi%29%2Asqrt%2810%29+=+20%28pi%29%2Asqrt%2810%29

And finally, then, the surface area of the frustum is

52%28pi%29%2B20%28pi%29%2Asqrt%2810%29