To solve our questions, we are going to use the kinematic equation for distance:
![d=vt](https://tex.z-dn.net/?f=d%3Dvt)
where
![d](https://tex.z-dn.net/?f=d)
is distance
![v](https://tex.z-dn.net/?f=v)
is speed
![t](https://tex.z-dn.net/?f=t)
is time
1. Let
![v_{w}](https://tex.z-dn.net/?f=v_%7Bw%7D)
be the speed of the wind,
![t_{w}](https://tex.z-dn.net/?f=t_%7Bw%7D)
be time of the westward trip, and
![t_{e}](https://tex.z-dn.net/?f=t_%7Be%7D)
the time of the eastward trip. We know from our problem that the distance between the cities is 2,400 miles, so
![d=2400](https://tex.z-dn.net/?f=d%3D2400)
. We also know that the speed of the plane is 450 mi/hr, so
![v=450](https://tex.z-dn.net/?f=v%3D450)
. Now we can use our equation the relate the unknown quantities with the quantities that we know:
<span>Going westward:
The plane is flying against the wind, so we need to subtract the speed of the wind form the speed of the plane:
</span>
![d=vt](https://tex.z-dn.net/?f=d%3Dvt)
![2400=(450-v_{w})t_{w}](https://tex.z-dn.net/?f=2400%3D%28450-v_%7Bw%7D%29t_%7Bw%7D)
Going eastward:
The plane is flying with the wind, so we need to add the speed of the wind to the speed of the plane:
![d=vt](https://tex.z-dn.net/?f=d%3Dvt)
We can conclude that you should complete the chart as follows:Going westward -Distance: 2400 Rate:
![450-v_w](https://tex.z-dn.net/?f=450-v_w)
Time:
![t_w](https://tex.z-dn.net/?f=t_w)
Going eastward -Distance: 2400 Rate:
![450+v_w](https://tex.z-dn.net/?f=450%2Bv_w)
Time:
2. Notice that we already have to equations:
Going westward:
![2400=(450-v_{w})t_{w}](https://tex.z-dn.net/?f=2400%3D%28450-v_%7Bw%7D%29t_%7Bw%7D)
equation(1)
Going eastward:
![2400=(450+v_{w})t_{e}](https://tex.z-dn.net/?f=2400%3D%28450%2Bv_%7Bw%7D%29t_%7Be%7D)
equation (2)
Let
![t_{t}](https://tex.z-dn.net/?f=t_%7Bt%7D)
be the time of the round trip. We know from our problem that the round trip takes 11 hours, so
![t_{t}=11](https://tex.z-dn.net/?f=t_%7Bt%7D%3D11)
, but we also know that the time round trip is the time of the westward trip plus the time of the eastward trip, so
![t_{t}=t_w+t_e](https://tex.z-dn.net/?f=t_%7Bt%7D%3Dt_w%2Bt_e%20)
. Using this equation we can express
![t_w](https://tex.z-dn.net/?f=t_w)
in terms of
![t_e](https://tex.z-dn.net/?f=t_e)
:
![t_{t}=t_w+t_e](https://tex.z-dn.net/?f=t_%7Bt%7D%3Dt_w%2Bt_e%20)
![11=t_w+t_e](https://tex.z-dn.net/?f=11%3Dt_w%2Bt_e)
equation
![t_w=11-t_e](https://tex.z-dn.net/?f=t_w%3D11-t_e)
equation (3)
Now, we can replace equation (3) in equation (1) to create a system of equations with two unknowns:
![2400=(450-v_{w})t_{w}](https://tex.z-dn.net/?f=2400%3D%28450-v_%7Bw%7D%29t_%7Bw%7D)
We can conclude that the system of equations that represent the situation if the round trip takes 11 hours is:![2400=(450-v_{w})(11-t_e)](https://tex.z-dn.net/?f=2400%3D%28450-v_%7Bw%7D%29%2811-t_e%29)
equation (1)
![2400=(450+v_{w})t_{e}](https://tex.z-dn.net/?f=2400%3D%28450%2Bv_%7Bw%7D%29t_%7Be%7D)
equation (2)
3. Lets solve our system of equations to find the speed of the wind:
![2400=(450-v_{w})(11-t_e)](https://tex.z-dn.net/?f=2400%3D%28450-v_%7Bw%7D%29%2811-t_e%29)
equation (1)
![2400=(450+v_{w})t_{e}](https://tex.z-dn.net/?f=2400%3D%28450%2Bv_%7Bw%7D%29t_%7Be%7D)
equation (2)
Step 1. Solve for
![t_{e}](https://tex.z-dn.net/?f=t_%7Be%7D)
in equation (2)
![2400=(450+v_{w})t_{e}](https://tex.z-dn.net/?f=2400%3D%28450%2Bv_%7Bw%7D%29t_%7Be%7D)
![t_{e}= \frac{2400}{450+v_{w}}](https://tex.z-dn.net/?f=t_%7Be%7D%3D%20%5Cfrac%7B2400%7D%7B450%2Bv_%7Bw%7D%7D%20)
equation (3)
Step 2. Replace equation (3) in equation (1) and solve for
![v_w](https://tex.z-dn.net/?f=v_w)
:
![2400=(450-v_{w})(11-t_e)](https://tex.z-dn.net/?f=2400%3D%28450-v_%7Bw%7D%29%2811-t_e%29)
![2400=(450-v_{w})(11-\frac{2400}{450+v_{w}} )](https://tex.z-dn.net/?f=2400%3D%28450-v_%7Bw%7D%29%2811-%5Cfrac%7B2400%7D%7B450%2Bv_%7Bw%7D%7D%20%29)
![2400=(450-v_{w})( \frac{4950+11v_w-2400}{450+v_{w}} )](https://tex.z-dn.net/?f=2400%3D%28450-v_%7Bw%7D%29%28%20%5Cfrac%7B4950%2B11v_w-2400%7D%7B450%2Bv_%7Bw%7D%7D%20%29)
![2400=(450-v_{w})( \frac{255011v_w}{450+v_{w}} )](https://tex.z-dn.net/?f=2400%3D%28450-v_%7Bw%7D%29%28%20%5Cfrac%7B255011v_w%7D%7B450%2Bv_%7Bw%7D%7D%20%29%20%20)
![2400= \frac{1147500+4950v_w-2550v_w-11(v_w)^2}{450+v_{w}}](https://tex.z-dn.net/?f=2400%3D%20%5Cfrac%7B1147500%2B4950v_w-2550v_w-11%28v_w%29%5E2%7D%7B450%2Bv_%7Bw%7D%7D%20)
![2400(450+v_{w})=1147500+2400v_w-11(v_w)^2](https://tex.z-dn.net/?f=2400%28450%2Bv_%7Bw%7D%29%3D1147500%2B2400v_w-11%28v_w%29%5E2%20)
![1080000+2400v_w=1147500+2400v_w-11(v_w)^2](https://tex.z-dn.net/?f=1080000%2B2400v_w%3D1147500%2B2400v_w-11%28v_w%29%5E2)
![(11v_w)^2-67500=0](https://tex.z-dn.net/?f=%2811v_w%29%5E2-67500%3D0)
![11(v_w)^2=67500](https://tex.z-dn.net/?f=11%28v_w%29%5E2%3D67500)
![(v_w)^2= \frac{67500}{11}](https://tex.z-dn.net/?f=%28v_w%29%5E2%3D%20%5Cfrac%7B67500%7D%7B11%7D%20)
![v_w= \sqrt{\frac{67500}{11}}](https://tex.z-dn.net/?f=v_w%3D%20%5Csqrt%7B%5Cfrac%7B67500%7D%7B11%7D%7D%20)
We can conclude that the speed of the wind is 78 mi/hr.