1. ## Modeling Equation.

I just don't understand how to start building equations, where to start. Plz Read the question below
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An aircraft flies 1062 miles with the wind at its tail. In the same
amount of time, a similar aircraft flies against the wind 738 miles. If the air speed
of each plane is 200 miles per hour, what is the speed of the wind? (Hint: Time
equals the distance divided by the speed.)
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The wind is in favor of the first air craft i.e. its pushing it and that's why it is able to cover more distance unlike the other air craft. so i can say that if x is the wind speed then for the first aircraft it is directly proportional so 200x, means its boosting the plane speed and for the other plane its 200/x as it is opposing it.

After this i go blank, now what. so plz help.

2. The speed at which an aircraft flies is the sum of the speed of the aircraft and the wind. For example, if the speed of the aircraft is 100 miles per hour and the speed of the wind is 50 miles per hour, then the speed at which the aircraft flies is 100 mph + 50 mph = 150 mph.

In your example, the first aircraft travels 1062 miles with the wind at its tail. Let $\displaystyle t$ be the time in which the first aircraft travels the 1062 miles. Then, the rate $\displaystyle r_1$ at which the first aircraft flies is

$\displaystyle r_1 = \dfrac{1062}{t}$

The rate $\displaystyle r_1$ at which the aircraft flies is the sum of the speed of the aircraft $\displaystyle v_a$ and the speed of the wind $\displaystyle v_w$. Therefore,

$\displaystyle v_a + v_w = \dfrac{1062}{t}$

The speed of the aircraft is given as 200 mph, so

$\displaystyle 200 + v_w = \dfrac{1062}{t}$ (1)

Similarly, the rate $\displaystyle r_2$ at which the second aircraft flies is

$\displaystyle 200 - v_w = \dfrac{738}{t}$ (2)

(Note the speed of the wind $\displaystyle v_w$ is subtracted from the speed of the second aircraft aircraft because the second aircraft flies against the wind.)

We have a system of equations (equations (1) and (2)), which can be solved for $\displaystyle v_w$ (the speed of the wind).

An aircraft flies 1062 miles with the wind at its tail.
In the same amount of time, a similar aircraft flies against the wind 738 miles.
If the air speed of each plane is 200 mph, what is the speed of the wind?

Let $\displaystyle \,w$ = speed of the wind (in mph).

The first plane flies 1062 miles at $\displaystyle 200\!+\!w$ mph.

. . This takes $\displaystyle \dfrac{1062}{200\!+\!w}$ hours.

The other plane flies 738 miles at $\displaystyle 200\!-\!w$ mph.

. . This takes $\displaystyle \dfrac{738}{200\!-\!w}$ hours.

The two times are equal: .$\displaystyle \dfrac{1062}{200 + w} \;=\;\dfrac{738}{200 - w}$

Solve for $\displaystyle \,w.$

4. Thanks Soroban. Would u kindly guide me for how to start modeling, some guide or some suggestions etc. thanks.