Let $d_1$ and $d_2$ represent the distances traveled at 90 km/h and 30 km/h, respectively. We are given the following:

• The total distance: $d_1 + d_2 = 150 \, \text{km}$.
• The total time: $t_1 + t_2 = 3 \, \text{hours}$, where $t_1$ is the time spent traveling at 90 km/h and $t_2$ is the time spent traveling at 30 km/h.

From the formula $\text{time} = \frac{\text{distance}}{\text{speed}}$, we know:

$t_1 = \frac{d_1}{90}, \quad t_2 = \frac{d_2}{30}$

We now have the system of equations:

1. $d_1 + d_2 = 150$
2. $\frac{d_1}{90} + \frac{d_2}{30} = 3$

Let's solve this system of equations step by step.

Starting with the second equation: $\frac{d_1}{90} + \frac{d_2}{30} = 3$

Multiply through by 90 to eliminate the denominators: $d_1 + 3d_2 = 270$

Now we have the system:

1. $d_1 + d_2 = 150$
2. $d_1 + 3d_2 = 270$

We can subtract the first equation from the second:

$(d_1 + 3d_2) - (d_1 + d_2) = 270 - 150$ $2d_2 = 120$ $d_2 = 60 \, \text{km}$

Substitute $d_2 = 60$ into the first equation:

$d_1 + 60 = 150$ $d_1 = 90 \, \text{km}$

Thus, the car traveled 90 km at 90 km/h and 60 km at 30 km/h.

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Would you like further details on this solution or have any questions?

Here are some related questions:

1. How do we find the time spent at each speed?
2. What is the average speed for the entire journey?
3. What would happen if the journey time was longer, but the distance stayed the same?
4. How can we adjust the formula if the car had multiple speed changes?
5. Can we model this using a graph of speed vs. time?

Tip: To find total time for a journey with multiple speeds, break the trip into segments based on speed and use the formula $\text{time} = \frac{\text{distance}}{\text{speed}}$.