Dalia flies an ultralight plane with a tailwind to a nearby town in [tex]$1 / 3$[/tex] of an hour. On the return trip, she travels the same distance in [tex]$3 / 5$[/tex] of an hour. What is the average rate of speed of the wind and the average rate of speed of the plane?
Initial trip:
Return trip:
Let [tex]$x$[/tex] be the average airspeed of the plane.
Let [tex]$y$[/tex] be the average wind speed.
Initial trip: [tex]$18=(x+y) \frac{1}{3}$[/tex]
Return trip: [tex]$18=(x-y) \frac{3}{5}$[/tex]
Dalia had an average airspeed of $\square$ miles per hour.
The average wind speed was $\square$ miles per hour.
Answer (2)
The average airspeed of the plane is 42 miles per hour, and the average wind speed is 12 miles per hour.
;
Set up a system of equations based on the given information: x + y = 54 and x − y = 30 , where x is the plane's airspeed and y is the wind speed.
Solve for x by adding the two equations: 2 x = 84 , which gives x = 42 .
Substitute the value of x into one of the equations to solve for y : 42 + y = 54 , which gives y = 12 .
The average airspeed of the plane is 42 miles per hour, and the average wind speed is 12 miles per hour.
Explanation
Problem Analysis Let's analyze the problem. We are given that Dalia flies to a nearby town with a tailwind in 3 1 of an hour and returns against the wind in 5 3 of an hour. We need to find the average airspeed of the plane and the average wind speed. We are given two equations:
Initial trip: 18 = ( x + y ) 3 1 Return trip: 18 = ( x − y ) 5 3
where x is the average airspeed of the plane and y is the average wind speed.
Simplifying the Equations First, let's simplify the given equations.
Equation 1: 18 = ( x + y ) 3 1 Multiply both sides by 3 to get: 54 = x + y
Equation 2: 18 = ( x − y ) 5 3 Multiply both sides by 3 5 to get: 30 = x − y
Solving for x Now we have a system of two linear equations with two variables:
x + y = 54 x − y = 30
We can solve this system by adding the two equations to eliminate y :
( x + y ) + ( x − y ) = 54 + 30 2 x = 84 x = 2 84 x = 42
Solving for y Now that we have the value of x , we can substitute it into either equation to solve for y . Let's use the first equation:
x + y = 54 42 + y = 54 y = 54 − 42 y = 12
Final Answer So, the average airspeed of the plane is 42 miles per hour, and the average wind speed is 12 miles per hour.
Examples
Understanding the effects of wind on travel is crucial in many real-world scenarios. For example, pilots need to calculate wind speed and direction to plan their routes efficiently and safely. Similarly, sailors and even long-distance runners need to account for wind resistance to optimize their performance. The math we used here, solving a system of equations, is a fundamental tool in these calculations, allowing for accurate predictions and adjustments to ensure successful journeys.
