A circular grass plot 40 m in radius is surrounded by a path. If the area of the grass plot is twice the area of the path, the width of the path in metres would be

Options

A. `40 (1+ sqrt(2/3))`
B. `40 (1 - sqrt(2/3))`
C. `40(sqrt(3/2) - 1)`
D. `40(sqrt(3/2) + 1)`

Answer

To solve the problem step by step, we need to find the width of the path surrounding the circular grass plot. Here's how to approach it:

### Step 1: Understand the Problem
We have a circular grass plot with a radius of 40 meters. The area of this grass plot is twice the area of the surrounding path. We need to find the width of the path, which we will denote as \( x \) meters.

### Step 2: Calculate the Area of the Grass Plot
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
For the grass plot with a radius of 40 meters:
\[ A_{\text{grass}} = \pi (40)^2 = 1600\pi \text{ square meters} \]

### Step 3: Express the Total Area Including the Path
The total radius of the circle including the path is \( 40 + x \) meters. Therefore, the area of the larger circle (grass plot + path) is:
\[ A_{\text{total}} = \pi (40 + x)^2 \]

### Step 4: Calculate the Area of the Path
The area of the path can be found by subtracting the area of the grass plot from the total area:
\[ A_{\text{path}} = A_{\text{total}} - A_{\text{grass}} = \pi (40 + x)^2 - 1600\pi \]
Factoring out \( \pi \):
\[ A_{\text{path}} = \pi \left((40 + x)^2 - 1600\right) \]

### Step 5: Set Up the Equation
According to the problem, the area of the grass plot is twice the area of the path:
\[ 1600\pi = 2 \cdot A_{\text{path}} \]
Substituting the expression for \( A_{\text{path}} \):
\[ 1600\pi = 2 \cdot \pi \left((40 + x)^2 - 1600\right) \]

### Step 6: Simplify the Equation
Dividing both sides by \( \pi \):
\[ 1600 = 2 \left((40 + x)^2 - 1600\right) \]
Expanding and simplifying:
\[ 1600 = 2(40 + x)^2 - 3200 \]
Adding 3200 to both sides:
\[ 4800 = 2(40 + x)^2 \]
Dividing by 2:
\[ 2400 = (40 + x)^2 \]

### Step 7: Solve for \( 40 + x \)
Taking the square root of both sides:
\[ 40 + x = \sqrt{2400} \]
Calculating \( \sqrt{2400} \):
\[ \sqrt{2400} = \sqrt{400 \times 6} = 20\sqrt{6} \]
Thus:
\[ 40 + x = 20\sqrt{6} \]

### Step 8: Solve for \( x \)
Subtracting 40 from both sides:
\[ x = 20\sqrt{6} - 40 \]

### Step 9: Final Calculation
To find the numerical value of \( x \):
\[ \sqrt{6} \approx 2.45 \Rightarrow 20\sqrt{6} \approx 49 \]
Thus:
\[ x \approx 49 - 40 = 9 \text{ meters} \]

### Final Answer
The width of the path is approximately \( 9 \) meters.

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