A telecommunications company is upgrading its network infrastructure to support increased demand for high-speed internet services. The upgrade requires installing fiber-optic cables along 500 miles of roads, at a cost of $100,000 per mile. In addition, the company needs to install signal amplifiers every 10 miles, which cost $50,000 each. Calculate the total cost of the network upgrade project, and estimate the time required to complete it, assuming a team of technicians can install 2 miles of cable and one amplifier per day.

First, let's calculate the cost of installing fiber-optic cables along 500 miles of roads. 

Cost of fiber-optic cable per mile = $100,000
Total miles = 500

Total cost for fiber-optic cables = Cost per mile * Total miles
Total cost for fiber-optic cables = $100,000 * 500 = $50,000,000

Next, let's calculate the number of signal amplifiers needed and their total cost.

Number of amplifiers = Total miles / Distance between amplifiers
Number of amplifiers = 500 miles / 10 miles/amplifier = 50 amplifiers

Cost of one amplifier = $50,000
Total cost for amplifiers = Number of amplifiers * Cost per amplifier
Total cost for amplifiers = 50 * $50,000 = $2,500,000

Now we can find the total cost of the network upgrade project by adding the costs of fiber-optic cables and amplifiers.

Total cost = Total cost for fiber-optic cables + Total cost for amplifiers
Total cost = $50,000,000 + $2,500,000 = $52,500,000

To estimate the time required to complete the project, we need to determine how many days it will take to install all the cables and amplifiers.

Each day, technicians can install:
- 2 miles of cable
- 1 amplifier

Since there are 500 miles of cable to install and technicians can install 2 miles per day:

Days required for cable installation = Total miles / Miles installed per day
Days required for cable installation = 500 miles / 2 miles/day = 250 days

Since there are 50 amplifiers to install and technicians can install 1 amplifier per day:

Days required for amplifier installation = Number of amplifiers / Amplifiers installed per day
Days required for amplifier installation = 50 amplifiers / 1 amplifier/day = 50 days

Assuming the technicians can work on both tasks simultaneously, the total time required to complete the project is equal to the longest of these two durations:

Time required for project completion = max(Days required for cable installation, Days required for amplifier installation)
Time required for project completion = max(250 days, 50 days) = 250 days