Question

A projectile is thrown upward so that its distance above the ground after t seconds is given by the function h(t) = -16t2 + 576t. After how many seconds does the projectile take to reach its maximum height?
(Show your work)

Answer 1

@myininaya would you mind helping with one last one??

Answer 2

it is asking you to find the vertex

Answer 3

actually it is asking you only for the x-coordinate (t-coordinate in this case) of the vertex

Answer 4

do you know how to write it in vertex form
or do you know the vertex formula

Answer 5

if not that is okay
we can work together to get there

Answer 6

Yeah, I'll need some help because I'm not sure

Answer 7

the vertex form of the parabola is
\[h(x)=a(x-h)^2+k \text{ where the vertex is } (h,k)\]
this if the form we want it in

Answer 8

first step factor the number that is in front of the squared variable out of the term with the squared variable and the variable
that is we are going to do this first:
\[h(t)=-16t^2+576t \\ h(t)=-16(t^2+\frac{576}{-16}t)\]
this is only the first step
I didn't simplify -576/16 yet because I couldn't do it in my head

Answer 9

that should be -36
\[h(t)=-16t^2+576t \\ h(t)=-16(t^2-36t)\]

Answer 10

now let's just look at t^2-36t....

Answer 11

this is the part we want to complete the square on

Answer 12

what is k here...
(go back to the first post we were in to determine this)

Answer 13

this thing
and here are some more examples
these examples are when the k is negative (like for our question)
\[x^2+kx+(\frac{k}{2})^2=(x+\frac{k}{2})^2 \\ x^2-2x+1=(x-1)^2 \\ x^2-4x+4=(x-2)^2 \\ x^2-6x+9=(x-3)^2 \\ x^2-8x+16=(x-4)^2 \\ x^2-10x+25=(x-5)^2 \\ x^2-12x+36=(x-6)^2 \\ x^2-14x+49=(x-7)^2\]

Answer 14

it doesn't matter t or x whatever

Answer 15

\[t^2-36t+... \text{ compare this to } t^2+kt+...\]
what does k need to be

Answer 16

36 ?

Answer 17

-36

Answer 18

oh, yeah, sorry

Answer 19

so this is good you determine what k needs to be
input that k into this formula:
\[t^2+kt+(\frac{k}{2})^2=(t+\frac{k}{2})^2 \]
everywhere you see k replace it with -36

Answer 20

\[t^2-36t+(\frac{-36}{2})^2=(t+\frac{-36}{2})^2\]
this what that should look like before simplifying

Answer 21

can you simplify

Answer 22

(t-18)^2

Answer 23

so you have
\[t^2-36t+(-18)^2=(t-18)^2 \\ \text{ or } t^2-36t+18^2=(t-18)^2 \\ \text{ or } t^2-36t+324=(t-18)^2\]

Answer 24

the first one?

Answer 25

oh no (that wasn't a multiple choice question just now... those 3 equations are all equivlanent... sorry )
ok this is going to help us with our problem
let me go up and fetch it...
this is what we left off with
\[h(t)=-16t^2+576t \\ h(t)=-16(t^2-36t) \]

anyways let's use the last simplified equation to help us complete the square here inside the parenthesis
you see we need to add 324 inside the parenthesis but this means we also need to subtract 324 inside the parenthesis

like this
\[h(t)=-16(t^2-36t+324-324) \\ \text{ now remember the distributive property } a(b+c)=ab+ac \\ \text{ I'm going \to do that here } \\ h(t)=-16(t^2-36t+324)-16(-324)\]
do you know why I did that?
or do you see what follows from doing this?

Answer 26

if you need a hint let me know

Answer 27

Just confused on where the 324 comes from

Answer 28

do you remember \[t^2-36t+(-18)^2=(t-18)^2 \\ \text{ or } t^2-36t+18^2=(t-18)^2 \\ \text{ or } t^2-36t+324=(t-18)^2\]

Answer 29

18^2 is 324 this is why we did the whole 324-324 into the parenthesis so we can complete the square

Answer 30

\[t^2+kt+(\frac{k}{2})^2=(t+\frac{k}{2})^2 \\ \text{ we had determined our } k \\ \text{ was } -36 \\ \text{ so we know the following equation is true } \\ t^2-36t+(\frac{-36}{2})^2=(t+\frac{-36}{2})^2 \]

Answer 31

-36/2 is -18
and (-18)^2 is the same as 18^2 which is the same as 324

Answer 32

so this is how we go this:
\[t^2-36t+324=(t-18)^2 \]

and that is where the 324 comes into play into our function

Answer 33

but whatever we add into our function we must also take out

Answer 34

\[h(t)=-16t^2+576t \\ h(t)=-16(t^2-36t) \\ h(t)=-16(t^2-36t+324-324) \\ \text{ we needed the 324 to complete the square } \\ \text{ so I just threw in +324 }\\ \text{ but also had to keep the function the same by putting in -324} \]

Answer 35

Okay, I get it up to this point now

Answer 36

\[h(t)=-16(\color{red}{t^2-36t+324}-\color{blue}{324}) \\ \text{ now the distributive property} \\ a(\color{red}{b}+\color{blue}c)=a \color{red}{b}+a \color{blue}{c} \\ h(t)=-16(\color{red}{t^2-36t+324})+-16(\color{blue}{-324})\]

Answer 37

guess what we can replace the t^2-36t+324 with ?

Answer 38

the whole point of adding the 324 was to complete the square

remember that we said t^2-36t+324 is the same as ....
( instead me saying the same as you might have seen the = which means the same thing )

Answer 39

(t-18)^2

Answer 40

yep yep yep!! :)

Answer 41

\[h(t)=-16(t-18)^2+-16(-324) \\ \]

Answer 42

we really don't need to simplify further to find the x-coordinate of the vertex

Answer 43

but if you want we can

Answer 44

-16(-324) is...

Answer 45

5,184

Answer 46

\[h(t)=-16(t-18)^2+5184 \\ \text{ compare this to } \\ h(t)=a(t-h)^2+k\]
the vertex is (h,k)

Answer 47

h will be your answer

Answer 48

whatever h is :)

Answer 49

18! so it takes 18 seconds for it to reach maximum height!

Answer 50

right on...
and the max height is 5184 (I know the question did not ask this)

Answer 51

Thank you so much

Answer 52

actually 5184 feet but whatever

Answer 53

no problem

Answer 54

I know that was really long
so give yourself a pat on the back for hanging in there that long
most people would give up
that should make your feel pretty successful right now

anyways peace