Airplane Algebra Calculator
This obviously can also be used for boats travelling upstream and dowmstream.
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 Algebra Problems concerning airplanes traveling with tailwinds, against headwinds, etc. are quite common (and quite tricky). You'll find this calculator quite helpful for solving such problems. Which information do you know?
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 Algebra problems concerning airplane velocity and headwind can take many forms. Here are 6 types of such problems. A) Solving for Airplane and Wind Velocity given Distance and Time Example: Travelling against the wind, an airplane takes 3 hours to travel 1,650 miles. On the return trip, the airplane travels with the wind, and takes 2 hours 45 minutes (or 2.75 hours) to travel 1,650 miles. What is the speed of the airplane in still air and the speed of the wind? Using the calculator, we click "A" then enter Distance   1650 Time 1   3 Time 2   2.75 (entering Time 1 = 2.75   Time 2 = 3 will also work) Clicking "Calculate" we see the answers are: Airplane Velocity   575 Wind Velocity     25

 Without using the calculator: Solving for both velocities Velocity (against the wind) = 1650 ÷ 3 = 550 miles per hour. Velocity (with the wind) = 1650 ÷ 2.75 = 600 miles per hour. "psa" means plane speed in still air; "ws" means wind speed psa -ws = 550 psa +ws = 600 Adding both equations we get 2*psa = 1150 Plane speed in still air = 575 Since psa +ws = 600 By substitution we get wind speed = 25 B) Solving for airplane velocity and distance given wind velocity and time. Example: An airplane flying with a 40 mph wind takes 4 hours to make a trip. On the return trip, the airplane flies against a 40 mph wind and takes 4.5 hours to make the trip. What is the airplane velocity and the distance travelled (one way)? Using the calculator, we click "B" then enter Wind Velocity   40 Time 1   4 Time 2   4.5 (entering Time 1 = 4.5   Time 2 = 4 will also work) Clicking "Calculate" we see the answers are: Airplane Velocity   680 Distance   2880 Without using the calculator: Distance = velocity * time Distance = (plane velocity + wind velocity) * time Dist (out) = (plane vel + 40) * 4 Dist (return) = (plane vel - 40) * 4.5 Since this is a round trip, the distance is the same so: (plane vel + 40) * 4 = (plane vel - 40) * 4.5 4*plane vel + 160 = 4.5 * plane vel - 180 340 = .5 * plane velocity plane velocity = 680 Putting this amount into this equation: Dist (out) = (plane vel + 40) * 4 Distance = (680 +40) * 4 Distance = 720 * 4 Distance = 2,880 C) Solving for Distance and Wind Velocity given airplane velocity and time. Example: You take a round-trip on an airplane, that has a velocity in still air of 440 mph. On your trip out, the plane flies for 6 hours against the wind and on your return trip, it flies for 5 hours with the wind. What is the wind speed and the one-way distance it traveled? Using the calculator, we click "C" then enter Airplane Velocity   440 Time 1   5 Time 2   6 (entering Time 1 = 6   Time 2 = 5 will also work) Clicking "Calculate" we see the answers are: Distance   2400 Wind Velocity   40 Without using the calculator, Distance = velocity * time Distance = (plane velocity + wind velocity) * time Distance (out) = (440 - wind velocity) * 6 Distance (return) = (440 + wind velocity) * 5 Since it's a round trip, the distance is the same so: (440 - wind velocity) * 6 = (440 + wind velocity) * 5 -6* wv + 2,640 = 2,200 +5*wv 440 = 11*wv wind velocity = 40 Putting this value into this equation: Distance (out) = (440 - wind velocity) * 6 Distance = (440 - 40) * 6 Distance = (400) * 6 Distance = 2,400 D) Solving for Wind Velocity given Airplane Velocity and Distance. Example: A plane's velocity in still air is 210 miles per hour. It flies for 725 miles with the wind and in the same amount of time, it flies 675 miles against the wind. What is the wind velocity? Using the calculator, we click "D" then enter Airplane Velocity   210 Distance 1   725 Distance 2   675 Entering Distance 1 = 675 and Distance 2 = 725 will also work Clicking "Calculate" we see the answer is: Wind Velocity   7.5 Without using the calculator There are 2 ways to do this. If you want the complete explanation (the solution that algebra teachers like), then just keep reading. Otherwise scroll to "the shorter method". We are not given a specific amount of time, but we do know that time = distance ÷ rate Since time is the same for both cases we can set up 2 "distance ÷ rate" fractions that equal each other. Distance1 ÷ (plane velocity + wind velocity) = Distance2 ÷ (plane velocity - wind velocity) 725 ÷ (210 + wind velocity) = 675 ÷ (210 - wind velocity) (725 ÷ 675) = (210 + wind velocity) ÷ (210 - wind velocity) (725 × 210) - (725 × wind velocity) = (675 × 210) + (675 × wind velocity) (725 × 210) - (675 × 210) = (725 × wind velocity) + (675 × wind velocity) 152,250 - 141,750 = 1,400 wind velocity 10,500 = 1,400 wind velocity wind velocity = 7.5 miles per hour The shorter method: When using this method, make sure distance1 is greater than distance2. We'll also abbreviate plane velocity and wind velocity as pv and wv. Insert the numbers from the problem into this equation: [(725 × 210) -(675 × 210)] ÷ (725 + 675) = wind velocity (152,250 -141,750) ÷ 1,400 = wind velocity 10,500 ÷ 1,400 = wind velocity wind velocity = 7.5 miles per hour E) Solving for Plane Velocity given Wind Velocity and Distance. Example: The wind velocity is 12 miles per hour. A plane flies for 850 miles with the wind and in the same amount of time, it flies 750 miles against the wind. What is the airplane velocity? Using the calculator, we click "E" then enter Wind Velocity   12 Distance 1   850 Distance 2   750 Entering Distance 1 = 750 and Distance 2 = 850 will also work Clicking "Calculate" we see the answer is: Airplane Velocity   192 Without using the calculator There are 2 ways to do this. If you want the complete explanation (the solution that algebra teachers like), then just keep reading. Otherwise scroll to "the shorter method". We are not given a specific amount of time, but we do know that time = distance ÷ rate Since time is the same for both cases we can set up 2 "distance ÷ rate" fractions that equal each other. Distance1 ÷ (plane velocity + wind velocity) = Distance2 ÷ (plane velocity - wind velocity) 850 ÷ (plane velocity + 12) = 750 ÷ (plane velocity - 12) (850 ÷ 750) = [(plane velocity + 12) ÷ (plane velocity - 12)] (850 × -12) + 850 pv = (750 × 12) + 750 pv -10,200 + 850 pv = 9,000 + 750 pv 100 pv = 19,200 plane velocity = 192 The shorter method: When using this method, make sure distance1 is greater than distance2. We'll also abbreviate plane velocity and wind velocity as pv and wv. Insert the numbers from the problem into this equation: [(850 × 12) + (750 × 12)] ÷ (850 -750) = plane velocity (10,200 + 9,000) ÷ 100 = plane velocity plane velocity = 192 F) Solving for Boat Velocity when given Current Velocity, Distance and Time. Example: Each day, a boat makes a 30 mile trip upstream and 30 miles downstream, taking 8 hours for the round trip. The velocity of the current is 5 miles per hour. What is the boat velocity? Using the calculator, we click "F" then enter Time   8 Distance   30 Wind (or Current) Velocity   5 Clicking "Calculate" we see the answer is: Airplane (or Boat) Velocity   10 Without using the calculator, We know that time = distance ÷ velocity. We only know the total time, so any equations we write must take that into account. Going upstream, the time would be: time1 = 30 ÷ (bv - 5) Going downstream, the time would be: time2 = 30 ÷ (bv + 5) One thing we do know is that the total time equals 8 hours. Since time1 + time2 = 8 hours, then we can say: [ 30 ÷ (bv - 5) ] + [ 30 ÷ (bv + 5) ] = 8 Multiplying both sides by (bv -5) 30 + [ 30 • (bv -5) ÷ (bv + 5) ] = 8bv -40 [ 30 • (bv -5) ÷ (bv + 5) ] = 8bv -70 Multiplying both sides by (bv +5) 30bv -150 = 8bv² -70bv +40bv -350 Collecting the terms 8bv² -60bv -200 = 0 Using the quadratic equation calculator boat velocity = 10 miles per hour
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