Heeeeelllllllllllppppppppppp!!!!!!!!!!!!!!!!!!?

Question

A spherical ball of r = 1/4 cm is placed in a corner touching both walls and floor. What is the radius of the largest ball which fits into the corner behind the spherical ball?

I need a convincing solution.

Answer 1

Think of a sphere of radius R inside a cube of side 2R. Then the distance from the center of the sphere (and cube) to one corner is (√3)R. So the distance from the sphere to the corner is (√3)R - R or R(√3 - 1). Likewise, the distance from the little sphere of radius r to the corner is r(√3 - 1). So from the center of the cube to the corner we have the radius of the large sphere and the diameter of the small sphere and the empty space. This gives us the formula:

(√3)R = R + 2r + r(√3 - 1)
R(√3 - 1) = r(2 + √3 - 1) = r(√3 + 1)
r = R(√3 - 1)/(√3 + 1)
r = R(2 - √3)

For R = 1/4 cm, the radius of the small sphere in the corner is:

r = (1/4)(2 - √3) = 0.067 cm

Answer 2

Just use your wild imagination...
Let A represents the corner, AB, AC and AD represent the intersections of the walls and floor, and O represents the center of the given spherical ball. The sphere is tangent to the walls and floor, therefore the distance of O to these surfaces is 1/4 cm. Let a cube of edge 1/2 cm circumscribe the ball.
OA is half the diagonal of the cube, which gives OA = sqrt(3) / 4 cm. Let S be the point on the surface of the sphere where OA passed, which is distant 1/4 cm from O. The center P, of the largest ball required lies on segment OA. The required ball is tangent to ball O at point S. Let r be the radius of the ball P. By the same method used in ball O, PA = sqrt(3)r.
OA = AP + PS + SO
sqrt(3) / 4 = sqrt(3)r + r + 1/4
r(sqrt(3)+1) = (sqrt(3)-1) / 4
Solving for r,
r = [sqrt(3) - 1] / [4(sqrt(3) + 1)] = (2 - sqrt(3)) / 4 cm.
Be convinced!!!

Answer 3

Unfortunately there is no scope to draw.
Treat the ball in 2D so that it becomes a circle touching the one wall and floor. The perpendicular distance from the centre of the circle to the wall = radius =1/4 cm. Similarly the perpendicular distance from the centre of the ball to the floor =radius = 1/4 cm. So the distance from the corner point to the centre of the circle = square root of summation of squares of 1/4 cm and 1/4 cm = square root of 1/16 cm +1/16 cm = square root of 1/8 cm = 0.3535. So the diameter of the largest spherical ball possible is 0.3535 - 0.25 = 0.1035 cm. Its radius thus is 0.1035/2 = 0.0518 cm

Answer 4

First look at the problem in 2 dimensions, it is the same problem but a little simpler.

Draw the corner (in this case 2 walls) Look at the square with the vertical and horizontal radii, and the extended radius from the center to the corner.

How far is it from the center of the circle to the corner? Clearly R*sqrt(2), so the segment from the circle to the corner is R*sqrt(2) - R.

This distance can also be computed from the little circle. It is r + r*sqrt(2), so (R*sqrt(2) - R) = r + r*sqrt(2)

r = ((R*sqrt(2) - R))/(1 + sqrt(2)).

In 3 dimensions it is solved the same way, but the pythagorean theorem has 3 terms rather than 2.

Answer 5

I'm quite sure that this is correct, but I will come back to find a way to show you my MS Paint on how I worked this out. Then I will add some explanations... Just wait there...

(√3/4)/(√3/4 - 1/4) = (√3/4 - 1/4)/√3/4 - 1/4 - 2r)
(√3)/(√3 - 1) = (√3 - 1)/(√3 - 1 - 8r)
3 - √3 - 8√3 r = 3 - 2√3 + 1
8√3 r = √3 - 1
24 r = 3 - √3
r = (3 - √3)/24 cm

^_^

Answer 6

Radius of the largest sphere which can fit into corner
=0.0726cm

Answer 7

Here is my solution:

radius of ball= .25cm
length of ground from centre of ball to corner - .25cm
=> distance from center of ball to corner = x^2 = .25^2 + .25^2
=>x^2 = 0.125
=> x = 0.354 (nearest decimal)

but the radius of the ball is .25. So subtract this from 0.354, and you have 0.104.

=> diameter of ball that can fit into the corner behind the big ball is 0.104, => the radius is 1/2 that, which is 0.052cm.

Answer 8

Take a drawing compass and set it to a radius of 1/4 cm. Draw a circle. Cut that circle out. Draw a right triangle on a piece of paper. Take the circle cut-out and line it up so that the perimeter is touching both sides of the right triangle. Using a ruler measure the space from the perimeter of the circle to the vertex of the triangle. That measurement is your diameter. Divide that by two and you have your radius.