Question

39. Three balls of equal radius are placed such that they are touching each other. A fourth smaller ball is kept such that it touches the other three. Find the ratio of the radius of larger to smaller ball.
a. (2-v3)/v3
b. 3-2×v3
c. 3.5
d. 4v5

I don't need just answer, I know which of them is correct. Please give a full solution of problem. Thanks

Answer 1

Answer:

The centers of three lower balls form an equilateral triangle of side  2r2r , where  rr  is the radius of the ball. Let C be the centroid of this triangle hence, by standard geometry, the distance of C from center of any lower ball(vertex of triangle) is  2r3–√2r3 . Also the center of the top(fourth) ball will lie exactly above C due to symmetry. Thus a right angled triangle is formed with centroid C, center of top ball, and center of lower ball. Using Pythagoras theorem, we can find the required height from C, and then add  rr  to this to find the net height from the ground.

h′=√(2r)2−(2r3–√)2

h′=2r√1−13

h′=2r√2/3

Thus is the height of center of 4th top ball from the plane in which centers of lower balls lie. To find net height from ground, add the radius of lower ball  rr  to  h′h′  

Height from ground,  h=h′+r

h=(2√2/3+1)

Hope it helps u !