Math Problem Statement

A plane flew 4000 km downwind in 6 hours, then 4000 km upwind in 8 hours. What is the speed of the plane and the speed of the wind?

Solution

The question reads:

A plane flew 4000 km downwind in 6 hours, then 4000 km upwind in 8 hours. What is the speed of the plane and the speed of the wind?

Let’s solve it step by step.


Variables:

  1. pp: Speed of the plane in still air (km/h)
  2. ww: Speed of the wind (km/h)

Downwind:

When flying downwind, the effective speed of the plane is: p+wp + w Using the formula: Speed=DistanceTime\text{Speed} = \frac{\text{Distance}}{\text{Time}} For the downwind leg: p+w=40006=666.6 km/hp + w = \frac{4000}{6} = 666.\overline{6} \text{ km/h}


Upwind:

When flying upwind, the effective speed of the plane is: pwp - w Similarly, for the upwind leg: pw=40008=500 km/hp - w = \frac{4000}{8} = 500 \text{ km/h}


Solve the System of Equations:

We now have two equations:

  1. p+w=666.6p + w = 666.\overline{6}
  2. pw=500p - w = 500

Adding the two equations: 2p=1166.62p = 1166.\overline{6} p=583.3 km/hp = 583.\overline{3} \text{ km/h}

Subtracting the second equation from the first: 2w=166.62w = 166.\overline{6} w=83.3 km/hw = 83.\overline{3} \text{ km/h}


Final Answer:

  • Speed of the plane (pp) = 583.3583.\overline{3} km/h
  • Speed of the wind (ww) = 83.383.\overline{3} km/h

Would you like a detailed explanation of any step or more practice problems?


5 Related Questions:

  1. What is the formula for effective speed when traveling with or against the wind?
  2. How would the solution change if the distances were different?
  3. What happens if the wind speed is greater than the plane's speed in still air?
  4. Can you derive a general formula for problems of this type?
  5. How does the time of travel affect these types of problems?

Tip:

Always double-check units and ensure consistency between time, distance, and speed when solving these problems!

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Distance-Speed-Time Relationship

Formulas

Speed = Distance / Time
p + w = Downwind Speed
p - w = Upwind Speed

Theorems

Linear equation solving

Suitable Grade Level

Grade 10