Question

A projectile is thrown upward so that its distance above the ground after t seconds is h = -16t2 + 672t. After how many seconds does it reach its maximum height?

Answer 1

so you have h = -16t2 + 672t
which is really a parabola, with a negative leading coefficient, meaning is a parabola going downwards

if we factor out the parabola form, to a vertex form of a parabola, we'd get the "vertex" of it, that is, its (h, k) or (x, y) values
how many "x" seconds it took to get to "y" height

Answer 2

that still doesn't make sense .-.

Answer 3

so let's start by getting common factor off it
h(t) = -16t2 + 672t => \(h(t) -16(t^2-42t)\)

now from there, we'd need a number to complete the "perfect trinomial"
that is a number at
\(\large h(t) -16(t^2-42t+\square?)\)

Answer 4

keep in mind that, to complete the square, all we're doing is using Mr zero 0 :)
that is, if we add say 200, we must also substract 200
so if we borrow 200, we must give back 200 to get 0 back

Answer 5

any number inside the parentheses, will be multiplied by -16, as you can see, thus the number will end up being a negative number, so we must add the quantity, that is
$$\large {
h(t) -16(t^2-42t+\square?)+(16\times \square?)\\
\text{so}\\
-(16\times \square?)+(16\times \square?)=0
}
$$

Answer 6

a perfect trinomial square has the form of
\(\large a^2\pm2ab+b^2\)
so our 2nd term, -42t is really => 2(ab), but we already know our "a = t" so
-42t => -2(t)(21)

so our 3rd number will be \(21^2\)

Answer 7

so
$$
h(t) -16(t^2-42t+21^2)+(16\times 21^2)\\
h(t)-16(t-\color{red}{21})^2+\color{red}{7056}\\
\text{so your vertex is at } (\color{red}{21}, \color{red}{7056})\\
$$
what does that mean?
well, after 21secs, the projectile is at 7056 feet