AS June 2018 Q3

EdexcelCurrent spec10 marksFlows in Networks

3.

Figure 1: directed network from S to T with capacities: SB 130, BA 40, BE 100, BC 60, AD 65, ED 10, ET 15, DT 75, EF 25, FE 30, CF 30, FT 85; cuts C1 and C2 shown as dashed lines
Figure 1

Figure 1 models the flow of fluid through a system of pipes from a source, S, to a sink, T.
The weights on the arcs show the capacities of the corresponding pipes in litres per minute.
Two cuts \(C_1\) and \(C_2\) are shown.

(a) Find the capacity of
(i) cut \(C_1\)
(ii) cut \(C_2\) (2)
(b) Using only the capacities of cuts \(C_1\) and \(C_2\) state what can be deduced about the maximum possible flow through the system. (1)
(c) On Diagram 1 in the answer book, show how a flow of 120 litres per minute from S to T can be achieved. You do not need to apply the labelling procedure to find this flow. (2)
Diagram 1: the directed network of Figure 1 with no values on the arcs
Diagram 1
(d) Prove that 120 litres per minute is the maximum possible flow through the system. (2)

A new pipe is planned from S to A. Let the capacity of this pipe be \(x\) litres per minute.

(e) Find, in terms of \(x\) where necessary, the maximum possible flow through the new system. (3)
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