Cost Projections: Tunneling

A company handling small tunnel-digging job in the Guadalupe Mountains in West Texas is in need of assistance in as much as their only staff mathematician, Mr. Mole, was murdered while putting the finishing touches on the job. He left only a page of sketchy notes concerning an important upcoming project. The company is requesting your help to finish where Mr. Mole had left off.

The cost of digging a tunnel depends on several factors, including length and diameter of the tunnel and then number of starting points for the tunnel. When one considers just the length when digging a tunnel, the cost per unit length is not constant: as the tunnel gets longer, the cost per unit length increases. This is due to the increasing expense of carrying tools and workers in, laying track and hauling dirt and rock out. This effect is summarized in the chart below which gives cost per foot estimates for digging a 50 foot diameter tunnel.

Length (ft.) 0-
100
100-
200
200-
300
300-
400
400-
500
500-
600
600-
700
700-
800
800-
900
900-
1000
Cost per foot ($/ft.) $500 $820 $1,180 $1,580 $2,020 $2,500 $3,020 $3,580 $4,180 $4,820

Your job is to answer the following questions concerning the assignment.

  1. How much would it cost to complete the digging of a one thousand foot tunnel with a diameter of 50 feet? Please provide a chart which show the total costs accrued at the 100 foot depth into the mountain, the 200 foot depth into the mountain, etc. up through the completion of the tunnel.
  2. How much money would they save (lose?) if they start this one thousand foot tunnel from both ends and have the two halves meet in the middle? While it is cheaper to have to dig only half-way, a supply factor of 1.35 has to been multiplied in to account for the need to maintain two separate bases for the dual operation.
  3. Using a graphing tool and the data of the cost per foot, find a "best-fit" line of the for the cost per foot function and a "best-fit" line for the total cost function for digging the tunnel as described in part 1. (HINT: The trend line of the graph of the cost per foot function is an exponential curve)