C. [tex]81\,\frac{9}{11}\,\frac{km}{h}[/tex]
Since we know the average speed, in kilometers per hour, associated to each stage of the trip, in kilometers, we can determine the average speed of the whole trip by means of the definition of weighted average:
[tex]\bar v = \frac{x_{1}\cdot v_{1}+x_{2}\cdot v_{2} + x_{3}\cdot v_{3}}{x_{1}+x_{2}+x_{3}}[/tex] (1)
Where:
- [tex]\bar v[/tex] - Average speed for the whole trip, in kilometers per hour.
- [tex]x_{1}, x_{2}, x_{3}[/tex] - Travelled distances for each stage, in kilometers.
- [tex]v_{1}, v_{2}, v_{3}[/tex] - Average speeds associated with each stage, in kilometers per hour.
If we know that [tex]x_{1} = 3\,km[/tex], [tex]v_{1} = 60\,\frac{km}{h}[/tex], [tex]x_{2} = 5\,km[/tex], [tex]v_{2} = 120\,\frac{km}{h}[/tex], [tex]x_{3} = 3\,km[/tex] and [tex]v_{3} = 40\,\frac{km}{h}[/tex], then the average speed for the whole trip is:
[tex]\bar v = \frac{(3\,km)\cdot \left(60\,\frac{km}{h} \right)+(5\,km)\cdot \left(120\,\frac{km}{h} \right)+(3\,km)\cdot \left(40\,\frac{km}{h} \right)}{3\,km + 5\,km + 3\,km}[/tex]
[tex]\bar v = 81.818\,\frac{km}{h}[/tex]
Hence, the correct choice is C.
We kindly invite to check this question on weighted averages: https://brainly.com/question/18554478
