A phone line runs east along a field for 1.5 miles and then north along the edge of the same field for 2.75 mile. If the phone line cost $3,500 per mile to install, how much could had been saved if the phone line had been installed diagonally across the field? How can I draw this In a form of a picture?
To find out how much could have been saved if the phone line had been installed diagonally across the field, let's start by drawing a picture:
1. Start by drawing a horizontal line to represent the eastward portion of the phone line. Label it as 1.5 miles.
2. From the endpoint of the horizontal line, draw a vertical line to represent the northward portion of the phone line. Label it as 2.75 miles.
3. Connect the starting point of the horizontal line with the endpoint of the vertical line using a diagonal line. This diagonal line represents the hypothetical route for the phone line if it had been installed diagonally across the field.
Now that we have the picture, we can calculate the lengths of the horizontal, vertical, and diagonal lines.
Using the Pythagorean theorem (a^2 + b^2 = c^2), where a and b are the lengths of the horizontal and vertical lines, and c is the length of the diagonal line, we can solve for c:
a = 1.5 miles
b = 2.75 miles
c^2 = 1.5^2 + 2.75^2
c^2 = 2.25 + 7.5625
c^2 = 9.8125
c = sqrt(9.8125)
c ≈ 3.13 miles (rounded to two decimal places)
Now that we know the length of the diagonal line (c ≈ 3.13 miles), we can calculate the cost savings:
The cost of installing a phone line is $3,500 per mile.
The cost of installing the original route (horizontal + vertical) is (1.5 + 2.75) * $3,500 = $18,750.
If the phone line had been installed diagonally across the field, the cost would have been 3.13 * $3,500 = $10,955.
Therefore, the potential cost savings would have been $18,750 - $10,955 = $7,795.
To visualize this situation, you can draw a diagram or use a grid system. Here's a simple way to represent the phone line's path:
|
2.75 miles
|
East A
1.5 miles ---------
| | /
| /
| /
| /
| /
| /
| /
In this diagram, the horizontal line going to the right represents the phone line running east for 1.5 miles. The vertical line going upwards represents the phone line running north along the edge for 2.75 miles. The point labeled "A" is the endpoint of the phone line. The diagonal line represents the direct path across the field that the phone line could have taken.
Now we can calculate the length of the diagonal line using the Pythagorean theorem.
The length of the diagonal line (d) can be found using the formula:
d^2 = (1.5 miles)^2 + (2.75 miles)^2
Simplifying this equation:
d^2 = 2.25 + 7.5625
d^2 = 9.8125
Taking the square root of both sides:
d = √9.8125
d ≈ 3.1305 miles
The phone line cost $3,500 per mile to install. So, to find out how much could have been saved, we calculate the cost difference between the actual installation and the diagonal installation.
Actual installation cost:
1.5 miles (east) = 1.5 miles * $3,500/mile = $5,250
2.75 miles (north) = 2.75 miles * $3,500/mile = $9,625
Total cost = $5,250 + $9,625 = $14,875
Diagonal installation cost:
3.1305 miles * $3,500/mile ≈ $10,952.75
Amount saved = Total cost - Diagonal installation cost = $14,875 - $10,952.75 ≈ $3,922.25
Therefore, approximately $3,922.25 could have been saved if the phone line had been installed diagonally across the field.