Question 1189898: An airplane travels 6282 kilometers against the wind in 9 hours and 7992 kilometers with the wind in the same amount of time. What is the rate of the plane in
stillair and what is the rate of the wind?
Found 4 solutions by Alan3354, math_tutor2020, MathTherapy, ikleyn:
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! An airplane travels 6282 kilometers against the wind in 9 hours and 7992 kilometers with the wind in the same amount of time. What is the rate of the plane in still air and what is the rate of the wind?
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Find the 2 groundspeeds, upwind and downwind.
The plane's airspeed is the average of the 2.
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Windspeed is the difference between airspeed and groundspeed.
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PS Knots are the units for airspeed, windspeed and groundspeed everywhere on Planet Earth, not km/hr.
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Not worth a Master's thesis, IMO.
Answer by math_tutor2020(3816)  (Show Source):
You can put this solution on YOUR website!
p = speed of the plane in still air (i.e. without any wind).
w = speed of the wind
speeds are in km per hour
Here are some useful terms to keep in mind- headwind = the wind that pushes against the head of the plane to slow it down. The wind is moving the opposite direction as the plane.
- tailwind = wind that pushes against the tail to speed the plane up. The wind is moving the same direction as the plane.
In short,- headwind = slows plane down
- tailwind = speeds plane up
Headwind:
When traveling against the wind, the headwind has the original plane speed (p) drop to p-w
distance = rate*time
distance = (p-w km/hr)*(9 hours)
distance = (p-w)*(9)
distance = 9(p-w)
We're told that the plane is able to travel 6282 km when encountering the headwind
So the first equation we can set up is
9(p-w) = 6282
Let's solve for p
9(p-w) = 6282
9(p-w)/9 = 6282/9
p-w = 698
p-w+w = 698+w
p = 698+w
Tailwind:
When traveling with the wind, the tailwind has the original plane speed (p) bump up to p+w
Through similar calculations as the headwind, we'll have...
distance = rate*time
7992 = (p+w)*9
From here, plug in p = 698+w and isolate w.
7992 = (p+w)*9
7992 = (698+w+w)*9
7992 = (698+2w)*9
7992 = 6282+18w
7992-6282 = 18w
1710 = 18w
18w = 1710
w = 1710/18
w = 95
The wind speed is 95 km per hour
Use that value of w to find p
p = 698+w
p = 698+95
p = 793
The plane travels at a speed of 793 km/hr if there wasn't any wind at all.
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Answers:
Plane speed in still air = 793 km per hour
wind speed = 95 km per hour
Answer by MathTherapy(10549)  (Show Source):
Answer by ikleyn(52750)  (Show Source):
You can put this solution on YOUR website! .
An airplane travels 6282 kilometers against the wind in 9 hours
and 7992 kilometers with the wind in the same amount of time.
What is the rate of the plane in still air and what is the rate of the wind?
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Let u be the airplane rate in still air (in kilometers per hour), and
let v be the rate of wind.
Then the airplane's effective speed flying with the wind is u+v kilometers per hour,
while its effective speed flying against the wind is u-v kilometers per hour.
From the condition, we have this system of equations
u + v = 7992/9 = 888 km/h (1)
u - v = 6282/9 = 698 km/h (2)
To solve the system, add the equations. You will get
2u = 888 + 698 = 1586; hence u = 1586/2 = 793 km/h.
Then from equation (1),
v = 888 - u = 888 - 793 = 95 km/h.
ANSWER. The rate of the airplane in still air is 793 kilometers per hour;
the rate of the wind is 95 kilometers per hour.
Solved.
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It is a typical "tailwind and headwind" word problem.
See the lessons
- Wind and Current problems
- Wind and Current problems solvable by quadratic equations
- Selected problems from the archive on a plane flying with and against the wind
in this site, where you will find other similar solved problems with detailed explanations.
Also, you have this free of charge online textbook in ALGEBRA-I in this site
ALGEBRA-I - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this textbook under the section "Word problems", the topic "Travel and Distance problems".
Save the link to this online textbook together with its description
Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson
to your archive and use it when it is needed.
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