A circular lawn has a path of uniform width around it. The area of the path is 21% of the area of the lawn. If the path is 7m wide, what is the radius of the lawn?

Let $r$ be the radius of the lawn.
The area of the lawn is $A_{lawn} = \pi r^2$.
The path has a uniform width of 7 m, so the outer radius (lawn + path) is $R = r + 7$.
The total area (lawn + path) is $A_{total} = \pi (r+7)^2$.
The area of the path is $A_{path} = A_{total} - A_{lawn} = \pi (r+7)^2 - \pi r^2$.

We are given that the area of the path is $21\%$ of the area of the lawn.

Multiply by 100 to clear decimals:

Divide by 7:

Using the quadratic formula $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

We get two possible values for $r$:

Since the radius cannot be negative, we take the positive value.

The radius of the lawn is $\boxed{\text{70 m}}$.

For this problem, we will use $\pi = \frac{22}{7}$ for exact integer results for the number of posts.
Inner radius $r = 70$ m.
Inner circumference $C_{inner} = 2 \pi r = 2 \times \frac{22}{7} \times 70 = 2 \times 22 \times 10 = 440$ m.
Outer radius $R = r + 7 = 70 + 7 = 77$ m.
Outer circumference $C_{outer} = 2 \pi R = 2 \times \frac{22}{7} \times 77 = 2 \times 22 \times 11 = 484$ m.

Posts on the inner side are at intervals of 10 m:

Posts on the outer side are at intervals of 11 m:

Total posts = $N_{inner} + N_{outer} = 44 + 44 = 88$ posts.

The total number of posts required is $\boxed{\text{88 posts}}$.

Total number of posts = 88.
Cost per post = sh 105.

Total cost = Total number of posts $\times$ Cost per post
Total cost = $88 \times 105 = 9240$ sh.

The total cost of the posts is $\boxed{\text{sh 9240}}$.