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When going to the beach, Guilherme takes 2 hours on the road to go from his city to the beach and 3 hours to get back to his city. Knowing that Guilherme's car goes at 100 km/h when downhill, 80 km/h on plane surfaces and 60 km/h  uphill, and that there are only 8 kilometers of road on plane surface, what is the distance in kilometers between the city and the beach?

 Jul 14, 2021
 #1
avatar+121065 
+1

Time spent on  plane  surface =   8km /  80km/h =  (1/10) h

 

 

Distance / rate   =  time

 

Let  x  be  the downhill part  and y  be  the uphill part  from the  city to the  beach

Then...on the return trip, the roles   are reversed.....y is the downhill part and x the uphill part

 

So   we  have  this system  of total times  ( in hrs)

 

x  / 100     + y/ 60  +   1/10  =    2

y/100  + x / 60   +  1/10   = 3           simplify

 

x/100 + y/60  =  1.9

y/100  + x/60  =  2.9               mutiply both  equations through  by  300.....arrange  as:

 

3x  +  5y  =  570     ⇒  mult through  by 5 ⇒      15x  + 25 y  =  2850

5x  +  3y   =  870    ⇒  mult through by -3 ⇒     -15x  - 9y  =  -2610               add these

 

16y  =  240

y =  15     (km)

 

3x  +  5(15)  =  570

3x  +  75  =  570

3x = 495

x = 165  (km)

 

Total  distance =    165  +  15 +  8  =  188 km

 

 

cool cool cool

 Jul 14, 2021
 #2
avatar+336 
+3

Hey there, Guest!

 

So...

 

Travelling at 8 Km at 80 km/hr takes 1/10 hours while travelling at \(X\) km at 100 km/hr takes \(\frac{x}{100}\) hours, and travelling at \(Y\) km at 60 km/hr takes \(\frac{y}{60}\) hours; in one direction, the sum of these is 2 hours, so \(\frac{1}{10}\) + \(\frac{x}{100}\) + \(\frac{y}{60}\) = 2. In the other direction, however, whatever was uphill becomes downhill, etc.; so the roles of x and y change, and we get the equation:

 

1/10 + x/60 +y/100 =3.

 

That equation can be simplified to get:

 

6x + 10y = 1140

&

6y + 10x = 1740

 

Therefore the answer is x=165 km, and y = 15 km.

So the total distance is 165 + 15 + 8 = 188 km.

 

Hope this helped! :)

( ゚д゚)つ Bye

 Jul 14, 2021
edited by TaliaArticula  Jul 14, 2021

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