ANSWER
The wind velocity is 43.2 miles
EXPLANATION
Given information
The total distance covered by the plane when flying with a tailwind is 550 miles
The total time = 2 hours
Let w represents the velocity of the wind
The speed with the wind = (160 + w)
The speed against the wind = (160 - w)
To find the wind velocity, follow the steps below
Step 1: Write the formula for writing distance
[tex]\text{ Speed = }\frac{distance\text{ }}{time}[/tex]Step 2: Find the time for the plane to travel with the wind
[tex]\begin{gathered} \text{ speed = }\frac{\text{ distance}}{time} \\ cross\text{ multiply} \\ distance\text{ = speed }\times\text{ time} \\ time\text{ = }\frac{distance}{speed} \\ time\text{ = }\frac{550}{(160\text{ + w\rparen}} \\ Hence,\text{ the time required for the plane to travel with wind is }\frac{550}{(160+\text{ w\rparen}} \end{gathered}[/tex]Step 3: Find the time for the plane to travel against the wind
[tex]\begin{gathered} \text{ speed = }\frac{distance}{time} \\ cross\text{ multiply} \\ distance\text{ = speed }\times time \\ time\text{ = }\frac{distance\text{ }}{speed} \\ time\text{ = }\frac{550}{(160-\text{ w\rparen}} \end{gathered}[/tex]Recall, that the total time is time against - time with
[tex]Time\text{ = Time against - time with}[/tex]Step 4:Substiute the value got in steps 2 and 3 into the formula in step 4
[tex]2\text{ = }\frac{550}{(160\text{ - w\rparen}}\text{ - }\frac{550}{(160+\text{ w\rparen}}[/tex]Step 5: Simplify the above expression
[tex]\begin{gathered} The\text{ common denominator = \lparen160 - w\rparen \lparen160 + w\rparen} \\ 2\text{ = }\frac{(160\text{ + w \rparen}\times550\text{ - \lparen160 - w\rparen}\times550}{(160-\text{ w\rparen\lparen160 + w\rparen}} \\ open\text{ the parentheses} \\ 2\text{ = }\frac{88000\text{ + 550w - 88000+ 550w}}{25600\text{ + 160w - 160w - w}^2} \\ collect\text{ the like terms} \\ 2\text{ = }\frac{88000\text{ - 88000 + 550w - 150 w}}{25600\text{ - w}^2} \\ 2=\text{ }\frac{1100w}{25600\text{ - w}^2} \\ cross\text{ multiply} \\ 1100w\text{ = 2\lparen25600 - w}^2) \\ 1100w\text{= 51200 - 2w}^2 \\ 1100w\text{ - 51200 + 2w}^2 \\ 2w^2\text{ +1100w -51200} \\ \end{gathered}[/tex]Step 6: Simplify the quadratic function using the general formula
[tex]\begin{gathered} 2w^2\text{ - 51200 +1100w} \\ \text{ Using, }\frac{-b\pm\sqrt{b^2\text{ - 4ac}}}{2a} \\ a\text{ = 2, b = 1100 c = -51200} \\ w\text{ = }\frac{-(1100)\pm\text{ }\sqrt{(1100^2\text{ - 4\lparen2 }\times\text{ -51200\rparen}}}{2\times2} \\ w\text{ = }\frac{-1100\pm\sqrt{1210000\text{ + 409600}}}{4} \\ w\text{ = }\frac{-1100\pm\sqrt{1619600}}{4} \\ w\text{ = }\frac{-1100\pm1272.635}{4} \\ \text{w = }\frac{-1100\text{ + 1272.635}}{4} \\ w\text{ = }\frac{172.635}{4} \\ w\text{ = 43.2 mile} \end{gathered}[/tex]Hence, the wind velocity is 43.2 miles
