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A right circular cone is sliced into four pieces by planes parallel to its base, as shown in the figure. All of these pieces have the same height. What is the ratio of the lateral surface area of the second largest piece to the lateral surface area of the largest piece? Express your answer as a common fraction.

Apr 17, 2024

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Analyze the Cone Slices:

Imagine the cone is cut into four slices with equal heights. Since the cuts are parallel to the base, each slice is a smaller similar cone.

Relate Lateral Surface Area to Slant Height:

The lateral surface area (LSA) of a cone is directly proportional to the slant height (l) of the cone. This means that the ratio of the lateral surface areas of two similar cones is equal to the ratio of their slant heights. Identify Slant Heights:

Let L be the slant height of the entire cone (the largest piece). The second-largest piece will have a slant height that is some fraction of L.

Analyze Proportion Based on Similar Triangles:

Since each slice is a similar cone, the ratio between the heights of the entire cone and the second-largest piece is the same as the ratio between their slant heights.

Looking at the slices, we can see that the height of the second-largest piece is 3/4 of the height of the entire cone. Therefore, the slant height of the second-largest piece (l') is also 3/4 of the slant height of the entire cone (L).

Calculate the Ratio of Lateral Surface Areas:

As mentioned earlier, the ratio of the lateral surface areas (LSA) is equal to the ratio of the slant heights:

LSA (second-largest piece) / LSA (largest piece) = l' / L

Substituting the values we found:

LSA (second-largest piece) / LSA (largest piece) = (3/4)L / L

Simplifying:

LSA (second-largest piece) / LSA (largest piece) = 3/4

Therefore, the ratio of the lateral surface area of the second-largest piece to the lateral surface area of the largest piece is 3:4, expressed as a common fraction.

Apr 17, 2024