A rectangle is drawn so that
the width is two inches
longer than the height.
If the rectangle's diagonal
measurement is 22 inches,
find the height and width.
Round to the tenths.
So looking at this rectangle here,
this is the diagonal which we know has
a length of 22 inches.
We also know the width
is two inches longer
than the height.
Let's use the variable h for the height
so this length here is h inches.
And because the width
is two inches longer,
we can use the expression
h plus two for the width.
Now from here notice how the diagonal cuts
the rectangle into two right triangles.
One here and one here.
And therefore we can use
the Pythagorean theorem
which c squared equals
a squared plus b squared
to determine the height and the width.
Where c is the length of the
hypotenuse, this length here,
and a and b are the
lengths of the two legs,
which are the two sides
that form the right angle.
So the diagonal must be c, the hypotenuse.
And then we can label
these two sides a and b.
It does not matter which
is a and which is b.
And now applying the Pythagorean theorem,
we would have c squared or 22 squared
equals a squared which
is equal to h squared
plus b squared which is the
quantity h plus two squared.
22 squared equals 484.
So we have 484 equals h squared plus,
there are no shortcuts
here, we have to write out
two factors of h plus
two and then multiply
which will give us four products.
We distribute the h,
then distribute the two.
H times h is h squared
so we have plus h squared
plus h times two is 2h, but notice how
the next product is two times
h which is also 2h giving
us plus 4h, and the
plus two times two which
gives us plus four.
On the right side we have two like terms,
h squared plus h squared is two h squared
giving us 484 equals two h
squared plus 4h plus four.
And now to set the equation equal to zero
we subtract 484 on both sides.
Simplifying, we now have
zero equals two h squared
plus 4h and then four minus
484 gives us minus 480.
Notice the greatest common factor
among all three terms is two.
Let's factor out the two from the right
which gives us zero equals two times
the quantity h squared plus 2h minus 240.
Let's continue on the next slide.
The trinomial inside the
parentheses does not factor,
and therefore we'll have to
use the quadratic formula
to determine the values of h that make
h squared plus 2h minus 240 equal to zero.
Well notice how a is equal to one,
b is equal to two, and c
is equal to negative 240.
So using the quadratic
formula shown here below,
we have h instead of x.
H equals negative b which is negative two
plus or minus the square root of b squared
which is two squared,
minus four times a times c
which is four times
one times negative 240.
All over two times a
which is two times one.
Simplifying we have h
equals negative two plus
or minus the square root
of this will be four,
of two squared which is four.
And then we have minus four
times one times negative 240
which is minus negative
960 which is equivalent
to plus 960, and all of this
is still divided by two.
Four plus 960 is equal to 964,
giving us h equals
negative two plus or minus
the square root of 964 all divided by two.
And now to approximate the two solutions
to the nearest tenth using the calculator.
For the first solution,
we have the quantity
negative two plus the square root of 964.
To get out from under the square root,
we press the right
arrow, close parentheses,
and then divided by two, which
gives us approximately 14.5.
And then for the second solution,
we can just press second enter
and then change the addition
sign to a subtraction sign
and press enter.
H is also approximately negative 16.5.
But in our case we can
exclude this solution
because we know h is a length.
So we now know h is approximately 14.5.
We can go back and determine the height
and the width.
So because the height is equal to h,
we know the height is
approximately 14.5 inches
and we know the width is
equal to h plus two inches.
Well 14.5 plus two is equal to 16.5
and therefore the width is
approximately 16.5 inches.
I hope you found this helpful.
