The speed of one of the two buses leaving from Izmir to Baku at the same time is 100 km/h. The fast bus arrives in Baku 3 hours before the slow one. How many km is the road between Izmir and Baku?

Let \( d \) be the distance between Izmir and Baku (in kilometers), and let \( v_s \) be the speed of the slower bus (in km/h).

The time it takes for the fast bus to travel this distance is:

\[
t_f = \frac{d}{100}
\]

The time it takes for the slower bus to travel the same distance is:

\[
t_s = \frac{d}{v_s}
\]

According to the problem, the fast bus arrives 3 hours before the slower bus, so:

\[
t_s = t_f + 3
\]

Substitute \( t_f \) and \( t_s \) into the equation:

\[
\frac{d}{v_s} = \frac{d}{100} + 3
\]

Now, multiply both sides by \( 100 \times v_s \) to eliminate the denominators:

\[
100d = v_s d + 300v_s
\]

Simplifying this:

\[
100d - v_s d = 300v_s
\]

Factor out \( d \) on the left-hand side:

\[
d(100 - v_s) = 300v_s
\]

Now solve for \( d \):

\[
d = \frac{300v_s}{100 - v_s}
\]

Thus, the distance \( d \) between Izmir and Baku depends on the speed \( v_s \) of the slower bus.

To find the exact value of \( d \), we would need to know \( v_s \). If you provide that information, I can continue with the calculation.

Would you like to proceed with an assumed value for \( v_s \), or do you have that information?

---

Relative questions:
1. What would change if both buses had different departure times?
2. How would the formula adjust if the fast bus had different speed variations?
3. Can you derive an equation for travel time differences with respect to speed changes?
4. How does the problem change if the difference in hours is 4 instead of 3?
5. Can you explore the relationship between speed and time in other transportation scenarios?

Tip: In problems involving relative motion, always express time in terms of distance and speed for each object before setting up equations.

Discover how to calculate the distance between Izmir and Baku with a detailed step-by-step solution. This problem involves key algebraic concepts and relative motion equations. Suitable for students in Grades 8-10.

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