5 marbles of various sizes are placed in a conical funnel as shown above. Each marble is in contact with adjacent marble(s). Also each marble is in contact all around the funnel wall. The smallest marble has radius 8mm and the largest has 18mm. What is the radius of the middle marble?
Answers
Answer:
The radius of the middle marble is 12mm
Consider two adjacent marbles, of radii a < b. We will show that b/a is a constant, whose value is dependent only upon the slope of the funnel wall.
The marbles are in contact with each other, and therefore the vertical distance between their centers is b + a.
The marbles are also in contact with the funnel wall. Since the slope of the funnel wall (in cross section) is a constant, the two green triangles are similar. Hence the horizontal distance from the center of each marble to the funnel wall is bc and ac, respectively, where c = sec(x) is a constant dependent upon the slope of the funnel wall. (x is the angle the funnel wall makes with the vertical.)
Let the slope of the funnel wall be m.
Then m = (b + a) / [(b − a)c].
Rearranging, b/a = (mc + 1)/(mc − 1).
Hence the ratio of the radii of adjacent marbles is a constant, dependent only upon the slope of the funnel wall. Let this constant be k.
In this case, we have 18 = 8k4.
So k2 = 3/2.
Therefore the radius of the middle marble is 8 · (3/2) = 12mm.
Remarks
Note that 12 is the geometric mean of 8 and 18. For any odd number of marbles in such a configuration, the radius of the middle marble is the geometric mean of the radii of the smallest and largest marbles.