Let's solve the problem step-by-step.
Given:
- The car traveled a distance of [tex]\( (2x + 13) \)[/tex] km at a speed of 67.5 km/h.
- The car then traveled a distance of [tex]\( (5x - 20) \)[/tex] km at a speed of 72 km/h.
- The total time for the journey is 90 minutes, which we need to convert to hours for consistency. This gives us [tex]\( \frac{90}{60} = 1.5 \)[/tex] hours.
### Step 1: Define the time taken for each part of the journey
1. Time to travel the first part:
The formula for time is:
[tex]\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\][/tex]
For the first part of the journey:
[tex]\[
t_1 = \frac{2x + 13}{67.5}
\][/tex]
2. Time to travel the second part:
For the second part of the journey:
[tex]\[
t_2 = \frac{5x - 20}{72}
\][/tex]
### Step 2: Set up the equation for total time
The total time for the journey is the sum of the times for each part, which should equal 1.5 hours:
[tex]\[
t_1 + t_2 = 1.5
\][/tex]
Substitute the expressions for [tex]\( t_1 \)[/tex] and [tex]\( t_2 \)[/tex]:
[tex]\[
\frac{2x + 13}{67.5} + \frac{5x - 20}{72} = 1.5
\][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex]
To solve for [tex]\( x \)[/tex], we need to clear the denominators. To do that, we can multiply the entire equation by the least common multiple (LCM) of 67.5 and 72. However, for simplicity, let's solve it directly:
Combining the fractions:
[tex]\[
\frac{2x + 13}{67.5} + \frac{5x - 20}{72} = 1.5
\][/tex]
We'll simplify the equation step by step to isolate [tex]\( x \)[/tex].
First, let's identify and combine like terms. The equation rewrites to:
[tex]\[
0.0296296296296296x + 0.192592592592593 + \frac{5x}{72} - \frac{5}{18} = 1.5
\][/tex]
Combining constants and terms involving [tex]\( x \)[/tex]:
[tex]\[
0.0296296296296296x + \frac{5x}{72} - 0.192593 + 0.49405266 = 1.5
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
0.0990740740740741x - 0.0851851851851852 = 1.5
\][/tex]
We then solve the equation:
[tex]\[
0.0990740740740741x - 0.0851851851851852 = 1.5
\][/tex]
Solving the above linear equation, we find:
[tex]\[
x = 16.0000000000000
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is:
[tex]\[
\boxed{16}
\][/tex]
