The ratio of the outer and the inner circumference of a circular path is 7 ∶ 6. If the path is 45 m wide, then what is the radius (in m) of the inner circle? 

Given:

The ratio of the outer circumference to the inner circumference of a circular path is 7:6.

The width of the path is 45 meters.

Formula Used:

Ratio of circumferences = Ratio of radii

Outer circumference = 2πR

Inner circumference = (2 π (r))

Where (R) is the radius of the outer circle and (r) is the radius of the inner circle.

Width of the path = (R - r)

Calculation:

Let the radius of the inner circle be (r) meters.

Then, the radius of the outer circle is (r + 45) meters.

Given the ratio of the outer circumference to the inner circumference is 7:6:

(2 π (r + 45))/(2 π r) = 7/6

(r + 45)/(r) = 7/6

Cross-multiply to solve for $r$ :

6 (r + 45) = 7r

6r + 270 = 7r

270 = 7r - 6r

270 = r

The radius of the inner circle is 270 meters.