A wooden pyramid is to be cut into 2 parts of equal weights by a plane parallel to its base. Find the ratio of the heights of the two parts.
To solve this problem, we can use the principle of similar triangles. Let's denote the original height of the pyramid as H and the ratio of the heights of the two parts as x:1 (where x is the height of the smaller part and 1 is the height of the larger part).
Since the ratio of the heights corresponds to the ratio of their weights, we can say that the weight of the smaller part (with height x) is to the weight of the larger part (with height 1) as x^3 is to 1^3.
Therefore, we have:
Weight of smaller part: Weight of larger part = x^3:1^3
Weight of smaller part: Weight of larger part = x^3:1
Since the two parts have equal weights, we can set up the equation:
x^3 = 1
Taking the cube root of both sides, we get:
x = 1^(1/3)
x = 1
So the height ratio is 1:1, meaning the two parts have equal heights.
In conclusion, the ratio of the heights of the two parts is 1:1.