An airplane has a speed of 400 km/h with no wind. The airplane flies 2140 km with the wind. The airplane can only fly 1860 km against the wind in the same time. If w equals the speed of the wind, make an equation that can be used to find w?
time = distance / rate ... the times are the same
2140 / (400 + w) = 1860 / (400 - w)
(2140 * 400) - 2140 w = (1860 * 400) + 1860 w
400 (2140 - 1860) = 4000 w
Sure, my friend! Let's break it down.
First, let's assume the speed of the wind is "w" km/h.
When the airplane flies with the wind, its effective speed is increased by the speed of the wind. So, the total speed of the airplane with the wind is (400 + w) km/h.
Similarly, when the airplane flies against the wind, its effective speed is reduced by the speed of the wind. So, the total speed of the airplane against the wind is (400 - w) km/h.
Now, we know that the airplane takes the same amount of time to fly both distances. Let's call this common time "t" hours.
The formula to find the time is given by the equation:
time = distance / speed
Using this formula, we can write two equations:
For flying with the wind:
t = 2140 / (400 + w)
For flying against the wind:
t = 1860 / (400 - w)
Since the time is the same in both cases, we can set these two equations equal to each other:
2140 / (400 + w) = 1860 / (400 - w)
Now, we can cross-multiply and simplify the equation to find "w". But I have a better idea! How about we let a math genius solve this equation, and I'll entertain you with a joke instead? Deal?
To find the speed of the wind, denoted as w, we can use the following equation:
(time taken flying with the wind) = (time taken flying against the wind)
Let's break down the problem step-by-step to construct the equation.
1. Find the time taken to travel 2140 km with the wind.
We know that the speed of the airplane with the wind is 400 km/h. Therefore, the time taken with the wind can be calculated by dividing the distance by the speed:
time_with_wind = distance / speed_with_wind
= 2140 km / 400 km/h
= 5.35 hours
2. Find the time taken to travel 1860 km against the wind.
We know that the speed of the airplane against the wind is 400 km/h - w (as the wind reduces the speed of the airplane). Therefore, the time taken against the wind can be calculated by dividing the distance by the speed:
time_against_wind = distance / speed_against_wind
= 1860 km / (400 km/h - w)
3. Set up the equation by equating the two times:
time_with_wind = time_against_wind
5.35 = 1860 km / (400 km/h - w)
Therefore, the equation that can be used to find the speed of the wind is:
5.35 = 1860 / (400 - w)
To solve this problem, we need to set up an equation based on the given information.
Let's define the variables:
- S = speed of the airplane in still air (without wind)
- w = speed of the wind
Given:
- Speed of the airplane with no wind = 400 km/h
- Distance traveled with the wind = 2140 km
- Distance traveled against the wind = 1860 km
When the airplane is flying with the wind, its effective speed is the sum of its own speed and the speed of the wind, i.e., S + w. So, the time taken to travel 2140 km can be calculated as distance divided by speed:
Time with the wind = Distance with the wind / (Speed of the airplane with no wind + Speed of the wind)
Time with the wind = 2140 km / (400 km/h + w)
Similarly, when the airplane is flying against the wind, its effective speed is the difference between its own speed and the speed of the wind, i.e., S - w. So, the time taken to travel 1860 km against the wind can be calculated as:
Time against the wind = Distance against the wind / (Speed of the airplane with no wind - Speed of the wind)
Time against the wind = 1860 km / (400 km/h - w)
Since the time taken for both scenarios is the same, we can set up an equation:
2140 km / (400 km/h + w) = 1860 km / (400 km/h - w)
This equation can be used to find the speed of the wind.