To save time a businessman
took a small airplane to
get to a lunch meeting
and then returned home.
The plane few a total of four hours
and each way the trip was 200 miles.
What was the speed of the
wind that affected the plane
which flies at a speed
of 120 miles per hour
with no wind?
To answer this question,
we will first use the formula
distance equals rate times time,
or d equals r times t,
and record the given information
in the table provided.
Once we have all the
information in the table,
we will set up and solve
a rational equation
in order to determine
the speed of the wind.
Let's also let the variable
w equal the wind speed,
and we will assume on the way there
there was a headwind, which
means a plane was flying
into the wind,
which means the rate of the plane would be
less than 120 miles per hour.
And on the way back there was a tailwind,
which means a plane was
flying with the wind,
and therefore the rate of the plane
is going to be more
than 120 miles per hour.
To begin, we know each way
the trip was 200 miles,
which means the plane few
200 miles with a headwind
and 200 miles with a tailwind.
And now let's talk about
the rate of the plane
with a headwind
and with a tailwind.
Again with a headwind,
the plane is flying into the wind,
and therefore the
overall rate of the plane
is going to be less
than 120 miles per hour,
which means the rate of the plane, r,
with a headwind is equal
to 120 miles per hour
minus the wind speed.
Let's record this in the table.
And when the plane is
flying with a tailwind,
again the wind is making
the plane go faster,
and therefore the rate of the plane, r,
is equal to 120 miles per hour
plus w, the wind speed.
Let's also record this in the table.
Now we don't know the time the plane flew
with a headwind
or the time the plane
flew with a tailwind.
But going back to the equation
distance equals rate times time,
if we solve the equation
for time, or the variable t,
we get time equals
distance divided by rate.
Because we have an expression
for the distance and rate,
we can write an expression
for the time the plane
flew with a tailwind
as well as a headwind.
The time the plane flew with a headwind
is equal to distance
divided by rate, or 200
divided by the quantity 120 minus w,
and the time the plane
flew with a tailwind
is equal to distance
divided by rate, or 200
divided by the quantity 120 plus w.
And then finally, we know the plane flew
for a total of four hours,
which means the time the
plane flew with a headwind
plus the time the plane
flew with a tailwind
must equal four hours.
Which gives us the rational equation 200
divided by the quantity 120 minus w
plus 200
divided by the quantity 120 plus w
must equal four.
This is the rational
equation we must solve for w
in order to determine the wind speed.
Let's solve this on the next slide.
To solve a rational equation,
we first clear the
fractions from the equation
by multiplying both sides of the equation
by the least common denominator.
Notice here the least common denominator
is the quantity 120 minus w times
the quantity 120 plus w.
Which means for the next step,
we'll multiply both sides of the equation
or all of the expressions on
both sides of the equation
by the product of the two denominators.
If it's helpful,
when the product involves a fraction,
we can write the product as a fraction
with a denominator of one, here and here.
And now we will determine each product,
but we will first simplify.
For the first product,
120 minus w divided by 120
minus w is equal to one.
The product simplifies to 200
times the quantity 120 plus w,
and then we have plus,
simplifying here 120 plus w
divided by 120 plus w is one,
giving us the product 200
times the quantity 120 minus w equals
on the right side nothing simplifies.
We have four times the
quantity 120 minus w
times the quantity 120 plus w.
And now to determine
each product on the left,
we distribute 200 here
as well as here.
200 times 120 is equal to 24,000,
and then we have plus 200 times w or 200w.
And we have plus 200 times 120,
again is 24,000,
and then we have minus 200 times w
which gives us minus 200w.
On the right side,
let's multiply the two binomials first,
which will give us four
times the quantity,
and then multiplying,
120 times 120 equals 14,400.
The next two products should be opposites.
Here we have positive 120w,
and then we have negative 120w,
so that gives us zero,
and then we have negative
w times positive w,
which gives us negative w
squared, or minus w squared.
Going back to the left
side of the equation,
we combine like terms.
We have two constant terms,
and we have two variable terms.
Looking at the variable terms,
200w minus 200w is zero.
Those two terms simplify out,
and 24,000 plus 24,000
equals 48,000.
So we have 48,000
equals on the right side,
we distribute four.
Four times 14,400 equals 57,600,
and we have minus four times w squared,
which gives us minus four w squared.
Notice how we have a quadratic equation,
but there's no w term,
only a w squared term.
Which means to solve the equation,
we will first isolate the w squared term
by subtracting 57,600
on both sides of the equation.
48,000 minus 57,600
equals negative 9,600.
On the right side,
this difference is zero,
we're left with negative four w squared.
The next step is to divide
both sides by negative four
to solve for w squared.
Negative 9,600 divided by negative four
equals positive 2,400.
On the right, negative four
divided by negative four
simplifies to one,
one times w squared is w squared.
And now to solve for w,
we need to undo the
squaring by square rooting
both sides of the equation.
Algebraically there is
going to be a positive
and negative solution,
but because we know w
represents wind speed,
we're only concerned about
the principle square root
or the positive square root.
So on the right side of the equation,
the square root of w
squared is equal to w.
And now to evaluate the
square root of 2,400,
we will use a calculator.
We press second x squared
for the square root,
enter 2,400,
press Enter.
Let's round to the nearest mile per hour
or the ones place value.
Because we have 48.9,
this would round to 49.
W is approximately 49.
And remember, w is the wind speed
or the speed of the wind
that affected the plane.
Which means we now know
the speed of the wind
that affected the plane
was approximately 49 miles per hour.
I hope you found this helpful.
