Question

The image shows a wooden stick of length 8 cm resting on two cylindrical rollers. The radius of the blue
and red rollers are 2 cm and 6 cm respectively. What is the distance (in cm) between the centres of the
rollers?​

Answer 1

The image shows a wooden stick of length 8 cm resting on two cylindrical rollers. The radius of the blue and red rollers are 2 cm and 6 cm respectively.

We have to find the distance between the centres of the rollers.

see the attached diagram, we drew a rough diagram on the same figure to understand more easily.

from ∆ABE and ∆ACD

∠AEB = ∠ADE = 90° [ tangents be perpendicular to radius ]

∠EAB = ∠DAC = θ [ common angle ]

hence, ∆EAB ~ ∆DAC

∴ EC/DC = AE/AD

⇒2cm/6cm = AE/8cm

⇒AE = 8/3 cm

so, DE = AD - AE = 8 - 8/3 = 16/3 cm

The distance between the centres of the rollers = BC

= $\sqrt{L^2+(r_2-r_1)^2}$

where L is the length of common tangent i.e., L = DE = 16/3 cm , r₁ = 2cm and r₂ = 6cm

= $\sqrt{\left(\frac{16}{3}\right)^2+(6-2)^2}$

= 6.67 cm

Therefore the distance between the centres of the rollers is 6.67 cm.