The height and feet of a projectile
is h equals negative
one-sixteenth d squared,
plus two d, plus five,
where d is the horizontal distance in feet
from the point at which
the object is projected.
A: What is the initial height
or the height of the
object when projected?
And B: when the projectile
first reaches a height
of nineteen feet, how far
has it traveled horizontally?
Let's first take a look
at this graphically.
The graph of the quadratic
function is shown here
in blue.
The initial height of the object
or the height of the object when projected
is the vertical intercept of
the graph at this point here.
The initial height or
the height of the object
when projected,
is when the horizontal
distance, d, traveled
is equal to zero.
Next, this horizontal line
is a graph of h equals 19.
Notice how it intercepts
the graph at two points.
These two points are when
the height of the object
reaches 19 feet,
and our goal is to determine
how far it's traveled
horizontally; at a height of 19 feet,
which occurs at d value
here, as well as here.
Because the questions asks,
"When the projectile first
reaches a height of 19 feet,
how far has it traveled horizontally?",
we only need to determine the d value
or the horizontal distance
traveled at this point here.
So going back to the question for A,
[mumbles] initial height,
or the height of the
object when projected,
we said d equaled to zero
and determined the value of h.
So when d is zero,
we need to determine the
function value h of zero,
which is negative one-sixteenth
times zero squared,
plus two times zero plus five;
this product is zero,
this product is zero,
and therefore h of zero equals five,
which means the initial height
of the object is five feet.
Now for B, to determine how
far the object has traveled
horizontally, when the
projectile first reaches a height
of 19 feet,
we said h equaled to zero
and solve for d.
So if h is equal to 19,
this gives us the equation:
19 equals negative
one-sixteenth d squared,
plus 2 d,
plus five.
Let's go ahead and solve this
equation on the next slide.
Because we have a quadratic equation,
let's enter equal to zero by
subtracting 19 on both sides.
This gives us zero equals
negative one-sixteenth d squared,
plus 2 d,
and then five minus 19
is equal to negative 14,
giving us minus 14.
For the next step,
let's clear the fraction
from the equation,
but instead of multiplying
both sides by 16,
let's multiply both sides by negative 16,
so the leading coefficient,
or the coefficient
of d squared is positive.
So we'll multiply both
sides by negative 16.
Again this will clear the
fraction from the equation.
On the left side, negative
16 times zero is zero;
and now distributing on the right,
negative 16 times negative
one-sixteenth d squared
is d squared.
Now we have negative
16 times positive 2 d,
that's negative 32 d,
giving us minus 32 d.
And then finally we have
negative 16 times negative 14,
which is positive 224, giving us plus 224.
Now we should try to solve by factoring,
but there are no factors of 224
that add to negative 32,
and therefore now need to
apply the quadratic formula,
where a is equal to one,
b is equal to negative 32,
and c is equal to positive 224.
Now applying the quadratic formula,
but in our case instead of x we have d.
We have d equals;
negative b is negative negative 32,
plus or minus the square root
of b squared, which is
the square of negative 32,
minus four times a times c,
which is minus four times one times 224.
All this is divided by two times a,
which is two times one.
Now we begin simplifying.
The opposite of negative 32,
or negative negative 32 is positive 32.
The square of negative 32 is 1,024.
Then we have minus four
times one times 224
equals eight hundred and ninety six.
This is all divided by two.
Now let's go ahead and
find this difference.
One thousand twenty four
minus eight hundred ninety six
equals one hundred and twenty eight.
The square root of 128 does
not simplify perfectly,
and therefore we will
now go to the calculator.
When entering in the calculator,
we need to make sure we have parentheses
around the numerator.
So we start with an open
parenthesis, and then 32;
let's first do the addition.
So plus, and then second x squared
brings up the square root.
We enter 128, press the right arrow
to get out from under the square root.
Close parenthesis, and
then divided by two.
Enter.
Running to the tenths place value,
notice how we have a six
in the tenths place value,
and then a five to the right,
which means you round up.
This is approximately 21.7.
So we have d is approximately
twenty one point seven, or;
and now we use a subtraction.
So going back to the calculator,
open parenthesis, 32
minus the square root,
128; right arrow, close
parenthesis divided by two.
Enter.
Running to the tenths place value,
we have a three in the tenths place value
and a four to the right.
We round down to 10.3.
The solutions to the equation are:
d is approximately 10.3,
or d is approximately 21.7.
But because the question reads,
"When the projectile first
reaches a height of 19 feet,
how far has it traveled horizontally?",
we use the smaller value of d,
which is approximately 10.3.
So now we know the projectile
has traveled 10.3 feet
horizontally when it first
reaches a height of 19 feet.
Let's go back and verify this graphically.
Here is where the object
reaches a height of 19 feet
for the first time,
and notice how the d value,
or the distance traveled horizontally,
does appear to be approximately 10.3.
I hope you found this helpful.
