Digging a tunnel The cost of digging a tunnel depends on many factors, including the overall length of the tunnel. However, the cost per unit length is not constant: as the tunnel gets longer, the cost per unit length increases because of the increasing expense of carrying tools and workers in and hauling dirt and rock out. Two companies bid to construct a 900 m tunnel. Consider the following diagram: Figure 1: Tunnel through hill. Their initial bids include details of how cost per metre changes with the overall length of the tunnel dug. For example, if a 400 metre tunnel was required, Company 1 would quote $1625 per metre and Company 2 would quote $1502 per metre. The quotes of cost per metre for tunnels ranging in length from 75 metres to 900 metres are summarised in the following table: Length of tunnel dug (m) 75 230 400 450 550 700 900 Cost per metre ($) Company 1 358 962 1625 1820 2210 2795 3575 Cost per metre ($) Company 2 300 873 1502 1687 2057 2612 3352 It is assumed that the relationship between cost per metre and length of tunnel dug is linear. As the project manager you investigate the relative costs of three different approaches to digging the tunnel: • use Company 1 only, • use Company 2 only, • use both Companies 1 and 2, each approaching from opposite ends of the tunnel and meeting at a location x metres from the western end of the tunnel. By using the following steps investigate which of the three options above will give you the minimum cost of constructing the tunnel. 1. Determine the function that relates cost per metre with length of the tunnel dug for Company 1 and use this function to determine the cost for Company 1 to dig the tunnel for this project alone. 2. Determine the function that relates cost per metre with length of the tunnel dug for Company 2 and use this function to determine the cost for Company 2 to dig the tunnel for this project alone. 3. Consider the situation where each company digs part of the tunnel, with each commencing at opposite ends, meeting at a point x metres from the western end of the tunnel. Use the two functions from (1) and (2) to develop a third function that relates total cost of digging the tunnel to the distance from the tunnel’s western end (x) and hence determine the total cost of digging the tunnel if the two companies share the digging, meeting at a point in the tunnel that will produce the minimum total cost for this option, (algebra and differential calculus must be used to determine the minimum total cost). If the companies share the task, determine how far from the western end of the tunnel the companies will meet when digging from opposite ends towards one another. Finally, conclude which of these three options would give minimum cost overall. Useful information: • The costs per metre for each company are the same no matter which end of the tunnel the digging is commenced from. • A single company cannot dig from both ends to meet in the middle because of time constraints and limitations on availability of equipment.