Answer to Question #153426 in Geometry for Reagan Forsythe

Question

A toy train set has a circular track peace the inner radius of the piece is 6 cm one sector of the truck has an arc length of 33 cm on the inside and 55 cm on the outside. What is the width of the track?

Answer 1

Let r₁ and r₂ be the radius of the inner circle and outer circle of the track respectively.

Let l₁ and l₂ be the length of the inner arc and outer arc of the sector of the track respectively.

Let $\theta$ be the angle of the sector.

Given:

r₁ = 6 cm

l₁ = 33 cm

l₂ = 55 cm

The length of an arc of a sector is given as:

l = ($\theta$/360) * 2$\pi$ * r

where r is radius and $\theta$ is sector angle.

Therefore,

l₁ = ($\theta$/360) * 2$\pi$ * r₁

33 = ($\theta$/360) * 2$\pi$ * 6

$\theta$ = (33 * 360)/(2$\pi$ * 6)

$\therefore$ $\theta$ = 315.12$\degree$

Similarly,

l₂ = ($\theta$/360) * 2$\pi$ * r₂

55 = (315.12/360) * 2$\pi$ * r₂

r₂ = (55 * 360)/(2$\pi$ * 315.12)

$\therefore$ r₂ = 10.0002 cm

i.e., r₂ $\approxeq$ 10 cm

Width of track = r₂ - r₁ = 10 - 6

$\therefore$ Width of track = 4 cm