Question 1077996: The speed of a sparrow is x km/h in still air. When the wind is blowing at 1 mph, the sparrow takes 5 hours to fly 12 miles to its nest and to return again. She flies out directly into the wind and returns with the wind behind her. Find her speed in still air. (Answer was 5 in the book)
Found 5 solutions by josgarithmetic, Theo, Alan3354, Fombitz, MathTherapy:
Answer by josgarithmetic(39617)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! x is her speed.
1 is the speed of the wind.
round trip is flying out and back again.
flying out, her speed is (x+1)
flying back, her speed is (x-1)
rate * time = distance.
let a = time it takes to get there.
let b = time it takes to get back.
total time taken was 5 hours.
a + b = 5
distance going is 12 miles.
distance coming back is 12 miles.
formulas of rate * time = distance become:
going with the wind against her:
(x-1) * a = 12
coming back with the wind behind her:
(x+1) * b = 12
since a + b = 5, you can solve for a to get a = 5 - b
formulas become:
going:
(x-1) * (5-b) = 12
coming back:
(x+1) * b = 12
you have reduced the number unknown variables to be solved down to 2 in 2 equations which means you can find a solution.
the 2 equations to be solved simultaneously are:
(x-1) * (5-b) = 12
(x+1) * b = 12
solve for b in the second equation to get b = 12/(x+1)
replace b with 12/(x+1) in the first equation to get:
(x-1) * (5-(12/(x+1)) = 12
now you're down to one unknown variable in one equation which means you can now solve for x.
there are a lot of messy calculations between here and the solution of x = 5.
i did them manually because it's easier to show what happens that way rather than trying to type them.
i'll show you them down further in this narrative.
suffice it to say that x = 5
this mean that the speed of the bird is equal to 5 km/hr.
when x = 5, the formula for the bird going is (5-1)*a = 12
simplify to get 4*a = 12
solve for a to get a = 12/4 = 3 hours.
when x = 5, the formula for the bird coming back is (5+1)*b = 12
simplify to get 6*b = 12
solve for b to get b = 12/6 = 2 hours.
a + b = 5 so this part of the problem statement is correct.
x = 5 is the solution you were given.
the messy calculations to get there from (x-1) * (5-(12/(x+1)) = 12 are shown below:
step 1 shows the equation to be solved.
step 2 multiplies both sides of the equation in step 1 by (x+1)
step 3 takes the middle part of the equation in step 2 and simplifies it.
step 4 then shows the equation from step 2 after it was simplified by the calculations in step 3.
step 5 simplifies the equation from step 4 further.
step 6 combines like terms of the equation in step 5.
step 7 subtracts (12x + 12) from both sides of the equation and gives you the result.
step 8 factors the equation in step 7.
step 9 shows the possible solutions for x.
step 10 shows the only valid solution for x.
i did not show how to solve the quadratic equation because you should be able to do so.
if you can't factor it directly, you can use the quadratic formula.
you will get the same answer i got if you did it correctly.
Answer by Alan3354(69443)  (Show Source):
Answer by Fombitz(32388)  (Show Source):
Answer by MathTherapy(10552)  (Show Source):
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