Question

During a tunnel construction project, engineers used explosives to excavate partway into a mountain. Because of time constraints and environmental concerns, they later used a tunnel boring machine (TBM) to excavate the rest of the tunnel. The table shows the length of the completed tunnel, y, after different numbers of days, x, since the TBM started working. Number of Days Length of Tunnel (feet) 0 225 5 350 20 725 30 975 55 1,600 The length of the completed tunnel can be calculated using the equation . If the project will be completed 86 days after the TBM was introduced, the length of the completed tunnel will be feet.

Answer 1

The given data can be tabulated as:

number of days,x                            length of tunnel (ft), y

0                                                                              225

5                                                                              350

20                                                                           725

30                                                                           975

55                                                                           1,600

 

To check if the growth is linear, the slope at points (0, 225) and (5, 350), and (0, 225) and (20, 725) should be equal. The formula for slope is:

m = (y2 – y1) / (x2 – x1)

 

Finding for slope (0, 225) and (5, 350):

m = (350 – 225) / (5 – 0) = 25

 

Finding for slope (0, 225) and (20, 725):

m = (725 – 225) / (20 – 0) = 25

 

Therefore the equation is linear in the form of:

y = mx + b

where b is the value of y at x = 0, so b = 225

 

The complete equation is:

y = 25x + 225

 

At x = 86 days

y = 25(86) + 225

y = 2,375 ft

 

Therefore the length of the completed tunnel is 2,375 feet.