The time, in minutes, it takes to travel by road between six towns is recorded and shown in the network diagram below.
- In this network the shortest path corresponds to the minimum travel time.
- What is the minimum travel time between towns \(A\) and \(F\), and what is the corresponding path? (2 marks)
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New roads are built to connect a town \(G\) to towns \(A\) and \(D\). The table gives the times it takes to travel by the new roads.
\begin{array} {|c|c|c|}
\hline
\rule{0pt}{2.5ex} \textit{Town} \rule[-1ex]{0pt}{0pt} & \textit{Time} \text{(minutes)} \rule[-1ex]{0pt}{0pt} & \textit{Town} \\
\hline
\rule{0pt}{2.5ex} A \rule[-1ex]{0pt}{0pt} & 8 \rule[-1ex]{0pt}{0pt} & G \\
\hline
\rule{0pt}{2.5ex} G \rule[-1ex]{0pt}{0pt} & 22 \rule[-1ex]{0pt}{0pt} & D \\
\hline
\end{array}
- Add the new roads and times to the network diagram below. (2 marks)
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- Explain whether the path in part (a) is still the shortest path from \(A\) to \(F\) after the new roads are added. (1 mark)
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a. \(\text{Minimum travel time}\ = 15+20+10+5+8=58\ \text{minutes}\)
\(\text{Path: }A → B → C → D → E → F\)
b. \(\text{New roads}\ A → G\ \text{and }G → D \)
c. \(\text{Using the new roads }A → G\ \text{and }G → D:\)
\(\text{Minimum travel time}\ =8+22+5+8=43\ \text{minutes.}\)
\(\text{Therefore the original path is no longer the shortest path.}\)
a. \(\text{Minimum travel time}\ = 15+20+10+5+8=58\ \text{minutes}\)
\(\text{Path: }A → B → C → D → E → F\)
b. \(\text{New roads}\ A → G\ \text{and }G → D \)
c. \(\text{Using the new roads }A → G\ \text{and }G → D:\)
\(\text{Minimum travel time}\ =8+22+5+8=43\ \text{minutes.}\)
\(\text{Therefore the original path is no longer the shortest path.}\)
