A plane traveled 580 miles to Anakara and back. The trip there was with the wind. It took 5 hours. The trip back was into the wind. The trip back took 10 hours. Find the speed of the plane in still air and the speed of the wind.
Well, it seems like this plane had a bit of a windy adventure! Let's call the speed of the plane in still air "P" and the speed of the wind "W".
When the plane was flying with the wind, it took 5 hours to cover 580 miles. So, the effective speed of the plane with the wind is P + W.
And when the plane was flying against the wind, it took 10 hours to cover the same distance. So, the effective speed of the plane against the wind is P - W.
Now, we know that distance equals speed multiplied by time. So, for each leg of the journey, we can form an equation.
For the journey with the wind, we have:
580 = (P + W) * 5
And for the journey against the wind, we have:
580 = (P - W) * 10
Now, let clown me crunch some numbers and find the solution for you!
Multiplying out the equations gives:
5P + 5W = 580
10P - 10W = 580
Adding the two equations together, we get:
15P = 1160
Dividing both sides by 15, we find:
P = 77.33
Now, let's substitute the value of P into one of the original equations to find W.
Using the first equation:
580 = (77.33 + W) * 5
Rearranging and solving for W, we get:
W = 580/5 - 77.33 = 115 - 77.33 = 37.67
So, the speed of the plane in still air is approximately 77.33 mph, and the speed of the wind is approximately 37.67 mph.
Let's assume the speed of the plane in still air is represented by p, and the speed of the wind is represented by w.
When flying with the wind, the effective speed of the plane is increased by the speed of the wind. Therefore, the speed of the plane with the wind is p + w.
When flying against the wind, the effective speed of the plane is reduced by the speed of the wind. Therefore, the speed of the plane against the wind is p - w.
Given that the distance traveled to Ankara and back is 580 miles, we can set up the following equations based on the given information:
Distance = Speed × Time
For the trip there:
580 = (p + w) × 5
For the trip back:
580 = (p - w) × 10
Now we can solve these equations simultaneously to find the values of p and w.
First, let's solve the equation for the trip there:
580 = 5p + 5w (equation 1)
Now, let's solve the equation for the trip back:
580 = 10p - 10w (equation 2)
We can rearrange equation 1 to isolate p:
5p = 580 - 5w
p = (580 - 5w) / 5
p = 116 - w (equation 3)
Substituting equation 3 into equation 2, we get:
580 = 10(116 - w) - 10w
580 = 1160 - 10w - 10w
580 = 1160 - 20w
20w = 1160 - 580
20w = 580
w = 580 / 20
w = 29
Now that we have the value of w, we can substitute it back into equation 3 to find p:
p = 116 - w
p = 116 - 29
p = 87
Therefore, the speed of the plane in still air is 87 mph and the speed of the wind is 29 mph.
To find the speed of the plane in still air and the speed of the wind, we can set up a system of equations based on the given information.
Let's assume the speed of the plane in still air is represented by "p" and the speed of the wind is represented by "w".
For the trip to Ankara (with the wind), the effective speed of the plane would be the sum of the speed of the plane in still air and the speed of the wind, so we have:
p + w = (distance) / (time)
p + w = 580 / 5
p + w = 116
For the trip back (against the wind), the effective speed of the plane would be the difference between the speed of the plane in still air and the speed of the wind, so we have:
p - w = (distance) / (time)
p - w = 580 / 10
p - w = 58
Now we have a system of equations:
p + w = 116 ...(Equation 1)
p - w = 58 ...(Equation 2)
We can solve this system of equations by either substitution or elimination method.
Let's use the elimination method. We'll add the two equations together:
(p + w) + (p - w) = 116 + 58
2p = 174
p = 174 / 2
p = 87
Now substitute the value of p back into one of the equations, let's use Equation 1:
87 + w = 116
w = 116 - 87
w = 29
So the speed of the plane in still air is 87 miles per hour and the speed of the wind is 29 miles per hour.
If the speed in still air is s, and the wind speed is w,
580/(s+w) = 5
580/(s-w) = 10
s=87
w=29