The circumferences of two circular paths are in ratio 2:7. If the smaller path has radius 8 m, what is the radius of the larger path?

The correct option is 28 m.

Step 1: Understand the given information
We are given two circular paths. Let:
- The radius of the smaller path be _r1 = 8 m.
- The radius of the larger path be _r2.
- The circumference of the smaller path be _C1.
- The circumference of the larger path be _C2.

The ratio of their circumferences is given as 2:7, which means:

$\frac{C_1}{C_2}=\frac{2}{7}$

Step 2: Use the formula for the circumference of a circle
The circumference of a circle with radius r is given by the formula:

$C=2\pi r$

Therefore, the ratio of the circumferences can be written as:

$\frac{2\pi r_1}{2\pi r_2}=\frac{2}{7}$

Canceling out the common term 2π from the numerator and denominator, we get:

$\frac{r_1}{r_2}=\frac{2}{7}$

This shows that the ratio of the radii of two circles is equal to the ratio of their circumferences.

Step 3: Substitute the known values and solve for the unknown radius
Substitute _r1 = 8 m into the equation:

$\frac{8}{r_2}=\frac{2}{7}$

Cross-multiply to solve for _r2:

$2\times r_2=8\times 7$

$2\times r_2=56$

$r_2=\frac{56}{2}$

$r_2=28\text{m}$

Thus, the radius of the larger circular path is 28 m.