We can solve this problem by applying the Pythagorean theorem to determine the diagonal distance across the field, and then compare the cost of installing the phone line along the edges to the cost of installing it diagonally.

Step 1: Calculate the diagonal length of the field

The Pythagorean theorem states that for a right triangle:

$\text{Diagonal}^2 = \text{East edge}^2 + \text{North edge}^2$

Here, the east edge is 2.8 miles and the north edge is 4.75 miles. So, we calculate the diagonal distance $d$ as follows:

$d = \sqrt{(2.8)^2 + (4.75)^2}$

$d = \sqrt{7.84 + 22.56}$

$d = \sqrt{30.4} \approx 5.52 \, \text{miles}$

Step 2: Calculate the cost for both cases

1. Cost of the current installation along the edges: The total distance is $2.8 + 4.75 = 7.55$ miles, and the cost per mile is $3100, so:

$\text{Total cost along the edges} = 7.55 \times 3100 = 23,405 \, \text{USD}$

2. Cost of installing the phone line diagonally: The diagonal distance is approximately 5.52 miles, so the cost would be:

$\text{Total cost diagonally} = 5.52 \times 3100 = 17,112 \, \text{USD}$

Step 3: Calculate the savings

The amount saved by installing the phone line diagonally instead of along the edges is:

$\text{Savings} = 23,405 - 17,112 = 6,293 \, \text{USD}$

Final Answer:

If the phone line had been installed diagonally across the field, approximately $6,293 could have been saved.

Do you want more details on any part of the explanation or have questions?

Here are five related questions to consider:

1. How is the Pythagorean theorem applied in real-world scenarios like this one?
2. What are other examples of cost-saving measures that involve distance optimization?
3. How does the cost per mile impact total savings in different situations?
4. Could weather or terrain make the diagonal installation impractical, even if it's cheaper?
5. How would savings change if the cost per mile for diagonal installation was different?

Tip: For right triangles, always remember the Pythagorean theorem to calculate the hypotenuse when you know the other two sides!