math problem, need help?

Question

AN uncrossed belt is fitted without slack around two circular wheels with radii 19cm and 3 cm. The distance between the initial points of contact of the belt with the wheels is 30 cm. Find the distance, in centimetres, between the centres of the wheels.

Answer 1

What you have here are two circles that are not touching, separated by a distance x, to be determined.

Radius big circle = R = 19 cm
Radius small circle = r = 3 cm

The distance between the initial points of contact is the distance between the two points of tangency of the line to the two circles. That distance is 30 cm.

Construct another circle inside the big circle and concentric to it with a radius equal to the difference in radius between the big and little circle (i.e. R - r = 19 - 3 = 16 cm). Call it the medium circle. If you construct a tangent line from the point at the center of the little circle to the medium circle, it will be parallel to the tangent line that is tangent to both the big and little circles. Furthermore, it will also be 30 cm in length.

Since a tangent line is perpendicular to the radius that meets it, we have a right triangle.

One leg is the medium radius R - r = 16 cm
The second leg is the tangent line from the center point of the small circle to the medium circle = 30 cm
The hypotenuse h, is the distance between the centers of the big and little circles.

h^2 = 16^2 + 30^2 = 256 + 900 = 1156
h = √1156 = 34 cm

Answer 2

8cm,unless I am missing something here.
19cm from ceter to edge and 3 cm from cether to edge = 22 total cm apart if the two were touching. The distance between to two wheels at the top (furthest most point) is 30 cm. Simply deduct 22 from 30.

Answer 3

The contact pts are at tangents. radius to tangent pt is perpindicular to tangent. we can construct a rt triangle with legs 16 & 30cm.
h^2=16^2+30^2=256+900=1156
h=34cm

Answer 4

4 cm,and 2 cm

Answer 5

30^2=16^2+x^2
x^2=30^2-16^2
=(30+16)(30-16)
=46*14
=644
x=rt644
=25.38cm