The length of a rectangular lot
is six feet less than
four times its width.
Its area is 216 square feet.
Determine the length and width.
So because the length is six feet less
than four times the width,
let's use a variable w
to represent the width,
this length here.
And then because the
length is six feet less
than four times its width,
we need an expression for
six less than four times w.
Well, six less than four
times w is the expression
four w minus six.
Again, four w is four times the width,
and then minus six gives us six less.
And the units are feet.
And we're told the area,
which is equal to length times
width, is 216 square feet.
Again, because the area of a rectangle
od equal to length times width,
we know in this case, 1216,
the area, must equal the length
of the quantity four w
minus six times the width,
which is w.
And now, let's clear the
parentheses on the right
by distributing w.
This gives us the equation
216 equals four w squared
minus six w.
Let's set the equation equal to zero
by subtracting 216 on both sides
which gives us zero equals four q squared
minus six w minus 216.
Remember, the first step in
factoring is to factor out
the greatest common factor.
On the right, the greatest
common factor is two,
giving us zero equals
two times the quantity
two w squared minus three w minus 108.
Now at this point, unfortunately,
the trinomial two w squared
minus three w minus 108
does not factor.
And therefore to determine the values of w
that make this trinomial
a quadratic equal to zero,
we will need to use the quadratic formula
shown here below.
Where a is the coefficient of w squared,
and therefore a is two.
B is the coefficient of w,
which is negative three,
and c is the constant
term of negative 108.
Notice how by factoring out
the greatest common factor
of two, we can use smaller
values for a, b, and c,
than using this form of
the quadratic equation.
And now, applying the quadratic formula,
we have w instead of x equals,
in the numerator, we have negative b,
which is negative of the
opposite of negative three
plus or minus the square
root of b squared,
which is the square of negative three,
minus four times a times c,
which is minus four times two
times negative 108.
And all this is divided by two times a,
which is two times two.
Now, let's begin simplifying.
We have w equals the
opposite of negative three
is Positive three.
Then we have plus or
minus the square root,
the square of negative
three is positive nine.
Then we have minus four
times two times negative 108
is equal to negative 864,
giving us minus negative 864.
And all this is divided by
two times two, which is four.
Subtracting a negative is
equivalent to adding a positive,
and therefore this
simplifies to nine plus 864,
which is equal to 873.
Which gives us w equals the
quantity three plus or minus
the square root of 873
all divided by four.
At this point, because
we're asked to round
to the nearest tenth,
we could go straight to the calculator,
but let's go ahead and simplify
the square root of 873.
873 is equal to nine times 97.
And because nine is a perfect square,
the square root of 873 is
equal to the square root
of nine times 97.
And because the square root
of nine is equal to three,
this simplifies to three square root 97.
So again, it's not required
because we're asked
to round to the nearest tenth,
but if we did want to simplify this,
we would have w equals
three plus or minus three
square root 97 all divided by four.
And now, let's go to the calculator
and get a decimal approximation
for the two solutions.
To evaluate this correctly,
I made a parentheses
around the numerator, so
we have open parenthesis
and then three plus three square root 97.
To get out from under the square root,
we press the right arrow.
Close parenthesis and
then divide it by four.
Rounding to the nearest tenth,
we have approximately 8.1.
Or I need to change
the sum in the numerator to a difference,
we can press second enter to ring up
the previous expression
and then just change the addition sign
to a subtraction sign.
Which gives us approximately negative 6.6.
Well, we know w is a length
and it can't be negative.
And therefore, we can
exclude this solution.
And now we know the width
is approximately 8.1 feet.
So again, we know w is
approximately 8.1 feet.
Now to find the length, we
need to determine the value
of the expression four w minus six
when w is approximately 8.1.
Which gives us four times 8.1 minus six,
which is equal to 26.4.
And of course, this is in feet.
So now we know the width
is approximately 8.1 feet.
And the length
is approximately 26.4 feet.
Before we go, let's verify a
rectangle with these dimensions
would have an area of
approximately 216 square feet.
Remember, area is equal
to length times width.
And therefore, the area is
going to be 26.4 times 8.1.
Which we can see is
approximately 216 square feet.
Just remember, we did round these values
to the nearest tenth, which is the reason
why we do have an error
when determining the area
using these dimensions.
I hope you found this helpful.
