Let's solve the problem step-by-step.
1. Define the given information:
- Distance traveled against the wind: 6860 km
- Time taken to travel against the wind: 7 hours
- Distance traveled with the wind: 7800 km
- Time taken to travel with the wind: 6 hours
2. Calculate the speed of the airplane against the wind:
We know that speed is distance divided by time. Therefore,
[tex]\[
\text{Speed against the wind} = \frac{\text{Distance against the wind}}{\text{Time against the wind}} = \frac{6860 \text{ km}}{7 \text{ hours}} = 980 \text{ km/h}
\][/tex]
3. Calculate the speed of the airplane with the wind:
Using the same speed formula,
[tex]\[
\text{Speed with the wind} = \frac{\text{Distance with the wind}}{\text{Time with the wind}} = \frac{7800 \text{ km}}{6 \text{ hours}} = 1300 \text{ km/h}
\][/tex]
4. Formulate the equations:
- Let \( p \) be the rate of the plane in still air (plane speed).
- Let \( w \) be the rate of the wind speed.
When flying against the wind:
[tex]\[
p - w = 980 \text{ km/h}
\][/tex]
When flying with the wind:
[tex]\[
p + w = 1300 \text{ km/h}
\][/tex]
5. Solve the system of equations:
- Add the two equations:
[tex]\[
(p - w) + (p + w) = 980 + 1300
\][/tex]
[tex]\[
2p = 2280
\][/tex]
[tex]\[
p = \frac{2280}{2} = 1140 \text{ km/h}
\][/tex]
- Subtract the first equation from the second:
[tex]\[
(p + w) - (p - w) = 1300 - 980
\][/tex]
[tex]\[
2w = 320
\][/tex]
[tex]\[
w = \frac{320}{2} = 160 \text{ km/h}
\][/tex]
6. Conclusion:
- The rate of the plane in still air: \( 1140 \text{ km/h} \)
- The rate of the wind: \( 160 \text{ km/h} \)
Therefore,
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Rate of the plane:} & 1140 \frac{\text{ km }}{\text{ h }} \\
\hline
\text{Rate of the wind:} & 160 \frac{\text{ km }}{\text{ h }} \\
\hline
\end{array}
\][/tex]
