Joe flew his small airplane 900 km in 5 hours flying with the wind. He flew 700 km against the wind in 7 hours. Find the rate at which he flew in still air and the rate of the wind... Joe flew his small airplane 900 km in 5 hours flying with the wind. He flew 700 km against the wind in 7 hours. Find the rate at which he flew in still air and the rate of the wind.
Understand the Problem
The question is asking to find the speed at which Joe flew in still air and the speed of the wind based on the distances he flew with and against the wind and the time taken for each journey.
Answer
The speed at which Joe flew in still air is \( 140 \) km/h, and the rate of the wind is \( 40 \) km/h.
Answer for screen readers
Joe flew at a speed of ( 140 ) km/h in still air, and the wind speed was ( 40 ) km/h.
Steps to Solve
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Define Variables Let ( x ) be the speed of the airplane in still air (km/h). Let ( y ) be the speed of the wind (km/h).
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Distance and Time with Wind When flying with the wind:
- Distance = 900 km
- Time = 5 hours
The equation for speed is: $$ x + y = \frac{900}{5} $$ This simplifies to: $$ x + y = 180 \quad \text{(Equation 1)} $$
- Distance and Time Against Wind When flying against the wind:
- Distance = 700 km
- Time = 7 hours
The equation for speed is: $$ x - y = \frac{700}{7} $$ This simplifies to: $$ x - y = 100 \quad \text{(Equation 2)} $$
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Solve the Equations Add Equation 1 and Equation 2: $$ (x + y) + (x - y) = 180 + 100 $$ This simplifies to: $$ 2x = 280 $$ Therefore, $$ x = 140 \text{ km/h} $$
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Find the Wind Speed Now substitute ( x ) back into Equation 1: $$ 140 + y = 180 $$ So, $$ y = 180 - 140 $$ Thus, $$ y = 40 \text{ km/h} $$
Joe flew at a speed of ( 140 ) km/h in still air, and the wind speed was ( 40 ) km/h.
More Information
This problem is a classic application of relative speed in which the speed of an object is affected by the movement of another object (in this case, the wind).
Tips
- Confusing the directions when setting up the equations. Remember that going with the wind adds speed, while going against it subtracts speed.
- Incorrectly calculating speeds from distances and times. Always ensure to divide the distance by the correct time.
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