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 3/5 of an hour. What is the average rate of speed of the wind and the average rate of speed of the plane?
Initial trip:
[tex]$3 / 5$[/tex] hours
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 [ ] miles per hour.
The average wind speed was [ ] miles per hour.
Answer (2)
Define variables x and y for the plane's airspeed and wind speed.
Set up equations based on the given information: ( x + y ) × 3 1 = 18 and ( x − y ) × 5 3 = 18 .
Solve for x + y and x − y to get x + y = 54 and x − y = 30 .
Solve the system of equations to find x = 42 and y = 12 . The average airspeed of the plane is 42 miles per hour, and the average wind speed is 12 miles per hour.
Explanation
Define variables and state the given information Let x be the average airspeed of the plane and y be the average wind speed. The distance to the nearby town is the same in both directions. When Dalia flies with the tailwind, her effective speed is x + y , and when she flies against the wind, her effective speed is x − y .
Write equations for the distances The distance to the town is the product of the effective speed and the time taken. Therefore, we have the following equations:
Distance with tailwind: d = ( x + y ) × 3 1 Distance against wind: d = ( x − y ) × 5 3
Equate the distances Since the distance is the same in both directions, we can equate the two expressions:
( x + y ) × 3 1 = ( x − y ) × 5 3
Solve for x+y We are also given that 18 = ( x + y ) × 3 1 and 18 = ( x − y ) × 5 3 . Let's solve the first equation for x + y :
18 = ( x + y ) × 3 1 ⟹ x + y = 18 × 3 = 54
Solve for x-y Now, let's solve the second equation for x − y :
18 = ( x − y ) × 5 3 ⟹ x − y = 18 × 3 5 = 6 × 5 = 30
Solve for x We now 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:
( x + y ) + ( x − y ) = 54 + 30 ⟹ 2 x = 84 ⟹ x = 42
Solve for y Now that we have the value of x , we can substitute it into one of the equations to find y . Let's use the first equation:
42 + y = 54 ⟹ y = 54 − 42 = 12
State the final answer Therefore, the average airspeed of the plane is 42 miles per hour, and the average wind speed is 12 miles per hour.
Examples
Understanding relative speeds is crucial in various real-life scenarios, such as navigation and logistics. For instance, pilots need to account for wind speed to accurately calculate flight times and fuel consumption. Similarly, boaters must consider the effect of currents on their speed and direction. This problem demonstrates how to calculate the average speed of a plane and the wind, which is essential for efficient travel and resource management.
Dalia's average airspeed in the ultralight plane is 42 miles per hour, while the average wind speed is 12 miles per hour. These values were found by setting up and solving a system of equations derived from the time and distance of her trips. The calculations reflect the effects of wind on her effective flying speed in both directions.
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