Answer:
The total duration of the trip is 48 hours.
Step-by-step explanation:
Let suppose that ship travels at constant speed during its travel. Each stage is represented by the following kinematic equation:
![v =\frac{\Delta s}{\Delta t}](https://tex.z-dn.net/?f=v%20%3D%5Cfrac%7B%5CDelta%20s%7D%7B%5CDelta%20t%7D)
Where:
- Travelled distance, measured in kilometers.
- Time, measured in hours.
- Speed, measured in kilometers per hour.
Now, each stage is represented by the following expressions:
Outbound trip
![v = \frac{700\,km}{\Delta t}](https://tex.z-dn.net/?f=v%20%3D%20%5Cfrac%7B700%5C%2Ckm%7D%7B%5CDelta%20t%7D)
Return trip
![v + 10\,\frac{km}{h} = \frac{700\,kh}{\Delta t - 8\,h}](https://tex.z-dn.net/?f=v%20%2B%2010%5C%2C%5Cfrac%7Bkm%7D%7Bh%7D%20%3D%20%5Cfrac%7B700%5C%2Ckh%7D%7B%5CDelta%20t%20-%208%5C%2Ch%7D)
By eliminating
and simplifying the resulting expression algebraically:
![\frac{700\,km}{\Delta t} + 10\,\frac{km}{h} = \frac{700\,km}{\Delta t -8\,h}](https://tex.z-dn.net/?f=%5Cfrac%7B700%5C%2Ckm%7D%7B%5CDelta%20t%7D%20%2B%2010%5C%2C%5Cfrac%7Bkm%7D%7Bh%7D%20%3D%20%5Cfrac%7B700%5C%2Ckm%7D%7B%5CDelta%20t%20-8%5C%2Ch%7D)
![(700\,km)\cdot \left(\frac{1}{\Delta t - 8\,h}-\frac{1}{\Delta t} \right) = 10\,\frac{km}{h}](https://tex.z-dn.net/?f=%28700%5C%2Ckm%29%5Ccdot%20%5Cleft%28%5Cfrac%7B1%7D%7B%5CDelta%20t%20-%208%5C%2Ch%7D-%5Cfrac%7B1%7D%7B%5CDelta%20t%7D%20%20%5Cright%29%20%3D%2010%5C%2C%5Cfrac%7Bkm%7D%7Bh%7D)
![\frac{1}{\Delta t - 8\,h}-\frac{1}{\Delta t} = \frac{1}{70}\,\frac{1}{h}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B%5CDelta%20t%20-%208%5C%2Ch%7D-%5Cfrac%7B1%7D%7B%5CDelta%20t%7D%20%3D%20%5Cfrac%7B1%7D%7B70%7D%5C%2C%5Cfrac%7B1%7D%7Bh%7D)
![\frac{8\,h}{\Delta t \cdot (\Delta t-8\,h)} = \frac{1}{70}\,\frac{1}{h}](https://tex.z-dn.net/?f=%5Cfrac%7B8%5C%2Ch%7D%7B%5CDelta%20t%20%5Ccdot%20%28%5CDelta%20t-8%5C%2Ch%29%7D%20%3D%20%5Cfrac%7B1%7D%7B70%7D%5C%2C%5Cfrac%7B1%7D%7Bh%7D)
![560\,h^{2} = \Delta t\cdot (\Delta t - 8\,h)](https://tex.z-dn.net/?f=560%5C%2Ch%5E%7B2%7D%20%3D%20%5CDelta%20t%5Ccdot%20%28%5CDelta%20t%20-%208%5C%2Ch%29)
![(\Delta t )^{2}-8\cdot \Delta t - 560 = 0](https://tex.z-dn.net/?f=%28%5CDelta%20t%20%29%5E%7B2%7D-8%5Ccdot%20%5CDelta%20t%20-%20560%20%3D%200)
This equation can be solved by means of the Quadratic Formula, whose roots are presented below:
and ![\Delta t_{2} = -20\,h](https://tex.z-dn.net/?f=%5CDelta%20t_%7B2%7D%20%3D%20-20%5C%2Ch)
Only the first roots offers a physically resonable solution. Then, total duration of the trip is:
![t_{T} = 28\,h +20\,h](https://tex.z-dn.net/?f=t_%7BT%7D%20%3D%2028%5C%2Ch%20%2B20%5C%2Ch)
![t_{T} = 48\,h](https://tex.z-dn.net/?f=t_%7BT%7D%20%3D%2048%5C%2Ch)
The total duration of the trip is 48 hours.