with a tailwind, a bird flew at a ground speed of 3 mi/h. Flying the same path against the same wind, the bird travels at a ground speed of 1.5 mi/h. What is the bird's air speed? What is the wind speed?
bird's air speed ---- x mph
wind speed -------- y mph
x+y = 3
x-y= 1.5
add them
2x = 4.5
x = 2.25 mph
y = 0.75 mph
To find the bird's air speed, we need to understand the effects of the wind on its ground speed. Let's assume that the bird's air speed is A mph and the wind speed is W mph.
When the bird is flying with a tailwind, the wind is assisting its movement, so its ground speed is the sum of its air speed and the wind speed. In this case, the bird's ground speed is 3 mi/h, so we can write the equation:
A + W = 3 (Equation 1)
When the bird is flying against the wind, the wind opposes its movement, so its ground speed is the difference between its air speed and the wind speed. In this case, the bird's ground speed is 1.5 mi/h, so we can write the equation:
A - W = 1.5 (Equation 2)
Now we have a system of two equations with two variables (A and W). We can solve this system of equations to find the values.
One way to eliminate a variable is to add both equations together. By adding Equation 1 and Equation 2, we get:
(A + W) + (A - W) = 3 + 1.5
This simplifies to:
2A = 4.5
Dividing both sides by 2, we find:
A = 2.25
Now that we know the bird's air speed is 2.25 mi/h, we can substitute this value into one of the original equations to solve for the wind speed.
Let's use Equation 1:
A + W = 3
Substituting A = 2.25, we get:
2.25 + W = 3
Subtracting 2.25 from both sides, we find:
W = 3 - 2.25
W = 0.75
Therefore, the bird's air speed is 2.25 mi/h, and the wind speed is 0.75 mi/h.
To solve this problem, we can set up a system of equations. Let's assume the bird's air speed is "a" (in mi/h) and the wind speed is "w" (in mi/h).
With a tailwind, the bird's ground speed is the sum of its air speed and the wind speed:
Ground speed with tailwind = Air speed + Wind speed = 3 mi/h
Against the wind, the bird's ground speed is the difference between its air speed and the wind speed:
Ground speed against the wind = Air speed - Wind speed = 1.5 mi/h
Now we have a system of equations:
a + w = 3 ...(Equation 1)
a - w = 1.5 ...(Equation 2)
To solve this system, we can use the method of elimination.
To eliminate "w", let's add Equation 1 and Equation 2:
(a + w) + (a - w) = 3 + 1.5
2a = 4.5
Dividing both sides of the equation by 2, we find:
a = 2.25 mi/h
Now, we can substitute the value of "a" into either Equation 1 or Equation 2 to solve for "w". Let's use Equation 1:
2.25 + w = 3
Subtracting 2.25 from both sides of the equation, we get:
w = 3 - 2.25
w = 0.75 mi/h
Therefore, the bird's air speed is 2.25 mi/h and the wind speed is 0.75 mi/h.