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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
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