A man running around a race course notes that the sum of the distances of two flagposts from him a always 10m and the distance between the flag posts is 8m. Then the area of the path he encloses in square meters is 15`pi` (b) `20pi` (c) `27pi` (d) `30pi`

Options

A. `15pi`
B. `12pi`
C. `18pi`
D. `8pi`

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the Problem
The problem states that a man running around a race course finds that the sum of the distances from two flagposts (let's call them F1 and F2) is always 10 meters. The distance between the two flagposts is 8 meters. This situation describes an ellipse, where the two flagposts are the foci.

### Step 2: Set Up the Parameters
In an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant. Here, that constant is 10 meters. Therefore, we can denote this constant as \( 2a = 10 \), where \( a \) is the semi-major axis.

- From \( 2a = 10 \), we find:
\[ a = \frac{10}{2} = 5 \text{ meters} \]

### Step 3: Find the Distance Between the Foci
The distance between the two flagposts (foci) is given as 8 meters. In terms of the ellipse, this distance is represented as \( 2c \), where \( c \) is the distance from the center of the ellipse to each focus.

- From \( 2c = 8 \), we find:
\[ c = \frac{8}{2} = 4 \text{ meters} \]

### Step 4: Use the Relationship Between a, b, and c
In an ellipse, the relationship between the semi-major axis \( a \), semi-minor axis \( b \), and the distance to the foci \( c \) is given by the equation:
\[ a^2 = b^2 + c^2 \]

### Step 5: Substitute Known Values
We already found \( a = 5 \) and \( c = 4 \). Now, we can substitute these values into the equation:
\[ 5^2 = b^2 + 4^2 \]
\[ 25 = b^2 + 16 \]

### Step 6: Solve for b
Now, we can isolate \( b^2 \):
\[ b^2 = 25 - 16 = 9 \]
Taking the square root gives:
\[ b = 3 \text{ meters} \]

### Step 7: Calculate the Area of the Ellipse
The area \( A \) of an ellipse is given by the formula:
\[ A = \pi \cdot a \cdot b \]
Substituting the values of \( a \) and \( b \):
\[ A = \pi \cdot 5 \cdot 3 = 15\pi \text{ square meters} \]

### Conclusion
The area of the path enclosed by the man running around the race course is \( 15\pi \) square meters.