xat 2025 Question 25

Question

The capital city, King&rsquo;s Landing, located 40 km to the south of Gulltown on the road connecting Gulltown to Winterfell, did not have a straight road, connecting to Meereen. Now, a new expressway is being built to connect these two towns by a straight road. What should be the maximum speed limit allowed on this expressway so that it cuts down the travel time, from Meeren to King&rsquo;s Landing, from the fastest possible route through the road network shown in the diagram, by 20 minutes?

Accepted Answer

We are told that a car always travelling at the maximum permissible speed, and always taking the shortest route takes 1 hour to reach Oldtown from Meereen, 2 hours to reach Gulltown from Oldtown, and 45 minutes to reach Winterfell from Gulltown. So, distance between Oldtown and Meereen is 100*1 = 100km Distance between Gulltown and Oldtown is 25*2 = 50km.

Distance between Winterfell and Gulltown. Is 120*3/4 = 90km.

Let the distance between Meereen and Winterfall be x.

Using the Pythagoras theorem, $(100+50)^2=90^2+x^2$

x = 120 km

For convenience, we are referring to the vertices as the starting letters of towns.

Using the similar triangle rule, MOL is similar to MGW.

So, $\frac{MO}{ML}=\frac{MG}{MW}$

$\frac{100}{ML}=\frac{150}{120}$

ML = 80 km.

LW = 40 km.

$\frac{MO}{LO}=\frac{MG}{WG}$

$\frac{100}{LO}=\frac{150}{90}$

LO = 60 km

The direct path from M to K will be $\sqrt{\ 120^2+50^2}=\ 130\ km$ .

Time taken for the path MOGK,= $\frac{100}{100}+\frac{50}{25}+\frac{40}{120}\ =1+2+\frac{1}{3}=3.33$

Time taken for the path MWK = $\frac{120}{120}+\frac{50}{120}\ =1.42$

So, the second case is MWK, which is the fastest route. The time taken will be 1 hr 25 mins.

If the travel time is cut down by 20 mins, the new time will be 1 hr 5 mins.

So, it should take 13/12 hrs. speed = $\dfrac{130}{\frac{13}{12}}=120 kmph$

Question 25.

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