Circular Lawn and Parallel Path: Solving Geometric Problems in Geometry

Geometry problems related to circles and paths, such as the one presented here, are common in mathematics. This problem involves a circular lawn surrounded by a path of uniform width, and the relationship between the area of the path and the lawn can be solved using algebraic methods. Let's delve into the step-by-step solution to find the radius of the circular lawn.

Problem Statement

A circular lawn is surrounded by a path of uniform width of 7 meters. The area of the path is 21 times that of the lawn. What is the radius of the lawn?

Step-by-Step Solution

Let's denote the radius of the lawn as r meters. The path surrounding the lawn has a uniform width, which means the radius of the entire area (lawn plus path) is r 7 meters.

Step 1: Calculate the Areas

- Area of the lawn:

```math A_{text{lawn}} pi r^2 ```

- Area of the entire region (lawn plus path):

```math A_{text{total}} pi (r 7)^2 ```

- Area of the path:

```math A_{text{path}} A_{text{total}} - A_{text{lawn}} pi (r 7)^2 - pi r^2 ```

- Simplifying the area of the path:

```math A_{text{path}} pi ( (r 7)^2 - r^2 ) pi ( r^2 14r 49 - r^2 ) ```

- Further simplification:

```math A_{text{path}} pi ( 14r 49 ) ```

Step 2: Relationship Between the Areas

According to the problem, the area of the path is 21 times the area of the lawn:

```math A_{text{path}} 0.21 A_{text{lawn}} ```

Substituting the expressions for the areas:

```math pi ( 14r 49 ) 0.21 pi r^2 ```

Step 3: Simplifying the Equation

We can cancel pi from both sides, as pi is not equal to zero:

```math 14r 49 0.21 r^2 ```

Step 4: Rearranging the Equation

Rearranging gives:

```math 0.21 r^2 - 14r - 49 0 ```

Step 5: Solving the Quadratic Equation

To solve the quadratic equation 0.21 r^2 - 14r - 49 0, we can use the quadratic formula:

```math r frac{-b pm sqrt{b^2 - 4ac}}{2a} ```

where a 0.21, b -14, c -49.

Calculating the Discriminant

Discriminant: b^2 - 4ac (-14)^2 - 4 cdot 0.21 cdot (-49)

```math b^2 - 4ac 196 41.16 237.16 ```

Now applying the quadratic formula:

```math r frac{14 pm sqrt{237.16}}{2 cdot 0.21} ```

Calculating sqrt{237.16} ≈ 15.4:

```math r frac{14 pm 15.4}{0.42} ```

Calculating the two potential values for r:

- r frac{14 15.4}{0.42} ≈ 70 - r frac{14 - 15.4}{0.42} (not a valid solution as radius cannot be negative)

Conclusion

The radius of the lawn is:

```math boxed{70 text{ meters}} ```

Therefore, the radius of the circular lawn is 70 meters.