Multiple choice

A cone hemisphere and a cylinder stands on equal bases and have the same height the height being equal to the radius of the circular base then their whole surface areas are in the ratio

  1. $ \displaystyle (\sqrt{2}+1):3:4 $
  2. $ \displaystyle (\sqrt{3}+1):3:4 $
  3. $ \displaystyle \sqrt{2}:3:4 $
  4. $ \displaystyle \sqrt{3}:7:8 $
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A Correct answer
Explanation

With radius r as the common height, the cone's whole surface area is (sqrt(2) + 1)pi r^2. The hemisphere and cylinder have whole surface areas 3pi r^2 and 4pi r^2, respectively. Their ratio is therefore (sqrt(2) + 1) : 3 : 4.

AI explanation

Let the common radius of the base be r, which also equals the height of all three solids. The total surface area of the cone is pi * r * (r + l), where the slant height l is r * square root of 2, making the area pi * r^2 * (square root of 2 + 1). The total surface area of the hemisphere is 3 * pi * r^2 and the total surface area of the cylinder is 2 * pi * r * (r + r), which is 4 * pi * r^2. Dividing these areas by pi * r^2 gives the ratio of (square root of 2 + 1) : 3 : 4.

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