Standard +0.8 This is a Chinese Postman Problem requiring identification of odd-degree vertices, calculation of minimum matching to make the graph Eulerian, comparison of total route length against fuel capacity constraints, and clear justification. While the individual steps are standard Further Maths techniques, the multi-stage application and real-world context modeling elevate it above routine exercises.
A company delivers parcels to houses in a village, using a van.
The network below shows the roads in the village. Each node represents a road junction and the weight of each arc represents the length, in miles, of the road between the junctions.
\includegraphics{figure_6}
The total length of all of the roads in the village is 31.4 miles.
On one particular day, the driver is due to make deliveries to at least one house on each road, so the van must travel along each road at least once. However, the driver has forgotten to add fuel to the van and it only has 4.5 litres of fuel to use to make its deliveries.
The van uses, on average, 1 litre of fuel to travel 7.8 miles along the roads of this village. Whilst making each delivery, the driver turns off the van's engine so it does not use any fuel.
Determine whether the van has enough fuel for the driver to make all of the deliveries to houses on each road of the village, starting and finishing at the same junction.
Fully justify your answer.
[6 marks]
Question 6:
6 | Sets up a model by identifying
the problem as a route
inspection problem and noting
that A, G, J and O are
odd-degree nodes (PI) | 3.3 | M1 | Odd degree nodes: A, G, J, O
Shortest Distances
A–G: 2.7 J–O: 2.0
A–J: 2.4 G–O: 2.5
A–O: 3.9 J–G: 2.4
Pairings
(A–G)(J–O) = 4.7*
(A–J)(G–O) = 4.9
(A–O)(J–G) = 6.3
Minimum total distance the van
must travel is 31.4 + 4.7 = 36.1
miles
The minimum amount of fuel the
van requires is
3 6 . 1
= 4.63 litres
7 . 8
Therefore, the van will require
more than 4.5 litres of fuel and so
does not have enough fuel to make
all of its deliveries and arrive back
at the junction it started from.
Uses the model to find at least
one correct total for a pair of
shortest distances | 3.4 | M1
Finds all three correct totals for
the pairs of shortest distances | 1.1b | A1
Determines their correct
minimum total distance that the
van needs cover during the
journey
or
Determines that the maximum
distance the van can travel with
4.5 litres of fuel is 35.1 miles | 1.1b | B1F
Determines the minimum
amount of fuel that would be
required
or
Makes a comparison of their
route with 35.1 miles | 1.1b | B1F
Uses the model to correctly
conclude that the van does not
have enough fuel to make all of
its deliveries and arrive back at
the junction it started from | 3.5a | E1F
Question total | 6
Q | Marking instructions | AO | Marks | Typical solution
A company delivers parcels to houses in a village, using a van.
The network below shows the roads in the village. Each node represents a road junction and the weight of each arc represents the length, in miles, of the road between the junctions.
\includegraphics{figure_6}
The total length of all of the roads in the village is 31.4 miles.
On one particular day, the driver is due to make deliveries to at least one house on each road, so the van must travel along each road at least once. However, the driver has forgotten to add fuel to the van and it only has 4.5 litres of fuel to use to make its deliveries.
The van uses, on average, 1 litre of fuel to travel 7.8 miles along the roads of this village. Whilst making each delivery, the driver turns off the van's engine so it does not use any fuel.
Determine whether the van has enough fuel for the driver to make all of the deliveries to houses on each road of the village, starting and finishing at the same junction.
Fully justify your answer.
[6 marks]
\hfill \mbox{\textit{AQA Further Paper 3 Discrete 2024 Q6 [6]}}