Question

A plane travels at a speed of 160 mph in still air. Flying with a tailwind, the plane is clocked over a distance of 550 miles. Flying against a headwind, it takes 2 hourslonger to complete the return trip. What was the wind velocity? (Round your answer to the nearest tenth.)AnswerKeypad

Answer 1

The wind velocity is 43.2 miles

Step-by-step 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

`\text{ Speed = }\frac{distance\text{ }}{time}`

Step 2: Find the time for the plane to travel with the wind

`\begin{gathered} \text{ speed = }\frac{\text{ distance}}{time} \\ cross\text{ multiply} \\ distance\text{ = speed }*\text{ time} \\ time\text{ = }(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}`

Step 3: Find the time for the plane to travel against the wind

`\begin{gathered} \text{ speed = }(distance)/(time) \\ cross\text{ multiply} \\ distance\text{ = speed }* time \\ time\text{ = }\frac{distance\text{ }}{speed} \\ time\text{ = }\frac{550}{(160-\text{ w\rparen}} \end{gathered}`

Recall, that the total time is time against - time with

`Time\text{ = Time against - time with}`

Step 4:Substiute the value got in steps 2 and 3 into the formula in step 4

`2\text{ = }\frac{550}{(160\text{ - w\rparen}}\text{ - }\frac{550}{(160+\text{ w\rparen}}`

Step 5: Simplify the above expression

Step 6: Simplify the quadratic function using the general formula

`\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 }*\text{ -51200\rparen}}}{2*2} \\ w\text{ = }\frac{-1100\pm\sqrt{1210000\text{ + 409600}}}{4} \\ w\text{ = }(-1100\pm√(1619600))/(4) \\ w\text{ = }(-1100\pm1272.635)/(4) \\ \text{w = }\frac{-1100\text{ + 1272.635}}{4} \\ w\text{ = }(172.635)/(4) \\ w\text{ = 43.2 mile} \end{gathered}`

Hence, the wind velocity is 43.2 miles