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CAT 2024 Slot 2 — DILR Question 4

Mixed PracticeEasy
Passage / Data

Answer the following questions based on the information given below.

Schematic of walkways and lakes

The above is a schematic diagram of walkways (indicated by all the straight-lines) and lakes (3 of them, each in the shape of rectangles – shaded in the diagram) of a gated area. Different points on the walkway are indicated by letters (A through P) with distances being OP = 150 m, ON = MN = 300 m, ML = 400 m, EL = 200 m, DE = 400 m.

The following additional information about the facilities in the area is known.
1. The only entry/exit point is at C.
2. There are many residences within the gated area; all of them are located on the path AH and ML with four of them being at A, H, M, and L.
3. The post office is located at P and the bank is located at B.

SegmentLength (m)SegmentLength (m)
AB, HG, IJ, PO150HI, GJ, FK, EL200
BC, CD300GF, FE300
JK, KL300ON, NM300
AH, IP, BG, JO400CF, KN, DE, LM400
GI (diagonal)250OK (diagonal)500

Coordinates (m), origin at P, taking the grid columns at x = 0, 150, 450, 750 and rows at y = 0, 400, 600, 1000: A(0,1000) B(150,1000) C(450,1000) D(750,1000); H(0,600) G(150,600) F(450,600) E(750,600); I(0,400) J(150,400) K(450,400) L(750,400); P(0,0) O(150,0) N(450,0) M(750,0). Lakes: C-D-E-F, G-F-K-J, K-L-M-N.

Visitors coming for morning walks are allowed to enter as long as they do not pass by any of the residences and do not cross any point (except C) more than once. What is the maximum distance (in m) that such a visitor can walk within the gated area?

Answer & solution

Answer: 3500

Solution

Medium

Residences sit on paths AH and ML, so the visitor must avoid points A, H, L, M (and segments AH, LM). Within the remaining points (B, C, D, E, F, G, I, J, K, N, O, P) find the longest simple closed walk through the entry point C.

Allowed points: B, C, D, E, F, G, I, J, K, N, O, P. Note P joins only O and I (so PO + PI are forced if P is used) and N joins only K and O. Avoid the short diagonals (GI 250, OK 500) when maximising.

1

Build the longest loop through C. Visit all 12 allowed points.

Loop: C-D-E-F-K-N-O-P-I-J-G-B-C\begin{aligned} &\text{Loop: } C\text{-}D\text{-}E\text{-}F\text{-}K\text{-}N\text{-}O\text{-}P\text{-}I\text{-}J\text{-}G\text{-}B\text{-}C \end{aligned}
2

Add the segment lengths.

CD+DE+EF+FK+KN+NO+OP+PI+IJ+JG+GB+BC⇒ 300+400+300+200+400+300+150+400+150+200+400+300⇒ 3500\begin{aligned} &CD{+}DE{+}EF{+}FK{+}KN{+}NO{+}OP{+}PI{+}IJ{+}JG{+}GB{+}BC\\ &\Rightarrow\ 300{+}400{+}300{+}200{+}400{+}300{+}150{+}400{+}150{+}200{+}400{+}300\\ &\Rightarrow\ 3500 \end{aligned}
3500 m3500\ \text{m}
CAT 2024 Slot 2 DILR Q4: Visitors coming for morning walks are allowed to enter as long as they do not pass by any of the residences an — Solution | TheCATExam