High School

One road segment has speed limits of 50 km/h, 60 km/h, and 100 km/h for 52 kilometers.

With a car, it takes 42 minutes, but with a truck, it takes at least 46 minutes. The maximum speed for a truck in Finland is 80 km/h.

How long are the different speed limit segments?

Answer :

The lengths of the segments with different speed limits are:

x = 3.37 km (50 km/h segment)

y = 21.95 km (60 km/h segment)

z = 26.69 km (100 km/h segment)

Let's denote:

x = distance of the segment with a speed limit of 50 km/h

y = distance of the segment with a speed limit of 60 km/h

z = distance of the segment with a speed limit of 100 km/h

Given information:

Total distance: x + y + z = 52 kilometers

Time taken by the car: 42 minutes

= 40/60 = 0.7 hours

Time taken by the truck: at least 46 minutes = 46/60 = 0.7667 hours

Maximum speed of the truck: 80 km/h

For the car:

The time taken by the car for each segment is the distance divided by the speed:

[tex]\frac{x}{50} + \frac{y}{60} + \frac{z}{100} = 0.7[/tex] hours

For the truck:

The time taken by the truck, considering its maximum speed is 80 km/h, is:

[tex]\frac{x}{50} + \frac{y}{60} + \frac{z}{80} \geq 0.7667[/tex]hours

System of Equations:

x + y + z = 52 -------eq1

[tex]\frac{x}{50} + \frac{y}{60} + \frac{z}{100} = 0.7[/tex]---------- eq2

[tex]\frac{x}{50} + \frac{y}{60} + \frac{z}{80} \geq 0.7667[/tex] ---------- eq3

These equations will allow us to solve for the distances x, y, and z.

From eqn2

Multiply through by the least common multiple (LCM) of 50, 60, and 100 to eliminate the fractions:

300x + 250y + 150z = 10500 ------- eqn4

From eqn 3

Multiply through by the LCM of 50, 60, and 80:

240x + 200y + 150z = 9200.4 ---------- eqn 5

Subtract Equation 5 from Equation 4 to eliminate z

(300x + 250y + 150z) - (240x + 200y + 150z) = 10500 - 9200.4

Simplify:

60x + 50y = 1299.6

Divide by 10 to further simplify:

6x + 5y = 129.96 -------- Equation 6

From Equation 1, express z in terms of x and y

z = 52 - x - y

Substitute z into Equation 2

300x + 250y + 150(52 - x - y) = 10500

300x + 250y + 7800 - 150x - 150y = 10500

150x + 100y = 2700 --------- eqn 7

Solveing equations 6 and 7 simultaneously

From eq6

y = 25.99 - 1.2x -------- 8

Substitute into eqn 7

150x + 100(25.99 - 1.2x) = 2700

150x + 2599 - 120x = 2700

30x = 101

x = 101/30

x = 3.37 km

Substitute x = 3.37 into eqn 8

y = 25.99 - 1.2(3.37)

y = 21.95 km

Substitute these values back into Equation 1 to find z

z = 52 - 3.37 - 21.95 = 26.69 km