Sure, let's solve the problem step-by-step.
### Step 1: Define the distances and speeds
The car travels two segments:
- The first segment has a distance of [tex]\(2x + 13\)[/tex] km at a speed of [tex]\(67.5\)[/tex] km/h.
- The second segment has a distance of [tex]\(5x - 20\)[/tex] km at a speed of [tex]\(72\)[/tex] km/h.
### Step 2: Convert the total travel time to hours
The total travel time is given as 90 minutes. To use consistent units (hours), we need to convert 90 minutes to hours:
[tex]\[ 90 \, \text{minutes} = \frac{90}{60} \, \text{hours} = 1.5 \, \text{hours} \][/tex]
### Step 3: Set up the time equations for each segment
The time taken to travel each segment can be found using the formula:
[tex]\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \][/tex]
- Time for the first segment:
[tex]\[ \text{Time}_1 = \frac{2x + 13}{67.5} \, \text{hours} \][/tex]
- Time for the second segment:
[tex]\[ \text{Time}_2 = \frac{5x - 20}{72} \, \text{hours} \][/tex]
### Step 4: Set up the total time equation
The total time for the journey is the sum of the times for each segment:
[tex]\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 \][/tex]
[tex]\[ 1.5 = \frac{2x + 13}{67.5} + \frac{5x - 20}{72} \][/tex]
### Step 5: Solve the equation for [tex]\(x\)[/tex]
Combine the two fractions into a single equation:
[tex]\[ 1.5 = \frac{2x + 13}{67.5} + \frac{5x - 20}{72} \][/tex]
We solve this equation to find the value of [tex]\(x\)[/tex]:
The solution to the equation is:
[tex]\[ x = 16 \][/tex]
### Conclusion
So, the value of [tex]\( x \)[/tex] is [tex]\( 16 \)[/tex].
