Question

Word Problem: A ball is thrown upwards. The height h, measured in feet, attained by it after being in the air for t seconds follows the rule h= -16t^2+32t+50. From what height was the ball thrown upward?

Answer 1

set t to 0 and you'll have the answer

Answer 2

since at the moment it is thrown it is not yet in the air, and since time started counting as soon as the ball is thrown in the air, t=0 is the exact moment the ball is thrown.

Answer 3

and thus, the height that will be the result of that equation is the height at which it is thrown.

Answer 4

Lol. Lets work this out together. I understand this more with actual numbers.

Answer 5

ok

Answer 6

zimmah is right, this equation relates the time and height of the ball, so when you just throw the ball, it will be t=0

Answer 7

So plug in 0 for every t you see, and there you go.

Answer 8

\[h=-16*0^2+32*0+50 = ?\]

Answer 9

Yeah, Im understanding him. I just want to clarify it more.

So Im replacing all of t with 0. Im going to get 50 as an answer.

Answer 10

@zimmah h=50?

Answer 11

yes

Answer 12

it's good to ask when you need more clarification, the purpose of asking is to understand it completely. you can never ask too many questions.

Answer 13

Sweet.

So if the ball was at 30feet above the ground ... how would I set that up for a solution?

Answer 14

You know how to find the time at which the ball crosses the x-axis right?

Answer 15

Like for example if i would say \[ax^2+bx+c=0\] you would be able to solve that, right?

Answer 16

Yeah, isn't that the form for Quadratic Equation?

Answer 17

correct. so let's use that knowledge, with a light alteration, to find what we need to know

Answer 18

because we do not want to know when it hits the ground, we want to know when it hits a 'imaginary ground' so to speak, that is 30 feet above it, and runs paralel to it.

Answer 19

so we are going to just say the ground is 30 feet above itself by saying

\[ax^2+bx+c=30\]

or in your case \[30= -16t^2+32t+50\]

Answer 20

however there is a catch

Answer 21

the quadratic equation only works when the other side is 0. so we have to make it 0 again by moving the 30 to the other side first.

Answer 22

so it becomes -30 and goes to the other side so that we get \[-16t^2+32t+50-30=0\]

Answer 23

Awwwh !!!

So therefore 30 - 50 .

Answer 24

Wait. Wait. Let me solve. on paper. Ill tell you the answer I get.

Answer 25

ok, however make sure you use 50-30. (I may have confused you a bit by having the equation the other way around)

Answer 26

50-30 will give 20.

So now I have:

0 = -16t^2 + 32t + 20

Answer 27

Im going to do the Quadratic Equation now.

Answer 28

yes, that's right.

Answer 29

So i solved and got:

\[\frac{ -32+/- [48] }{ -32 } \]

Answer 30

oh hey

Answer 31

lets see

Answer 32

ye, we can even simplyfy it a bit more to two readable answers.

Answer 33

you know, the +/- means you get two answers, one for the+and one for the -

Answer 34

note that this not necessarily means you get a positive and a negative end results, it's just the number under the root that can be both positive and negative

Answer 35

\[\frac{ -32+48 }{ -32 }\] and \[\frac{ -32-48 }{ -32 }\]

Answer 36

I got these:

\[\frac{ 16 }{ -32 } and \frac{ -80 }{ -32 } = \frac{ 80 }{ 32 }\]

Answer 37

yes, these numbers are correct still, yes you can still simplify them.

Answer 38

you can easily tell if you can simplify a number by breaking it into prime numbers. Prime numbers are the 'atoms' of the numbers so to speak. It's pretty interesting because every number has a unique signature of prime numbers.

Answer 39

usually it's enough to know the first few, 2,3,5,7,9,11,13 you just try to divide until you can no longer divide and eventually you will end up with the DNA of the number

Answer 40

now why would you want to do that, you may ask?

Answer 41

\[\frac{ 16 }{ -32 }\frac{ 2*2*2*2 }{ -2*2*2*2*2 }=\frac{ 2^4 }{ -2^5 }=\frac{ 1 }{ 2 }\]

Answer 42

I simplified all the way and got this:

\[-\frac{ 16 }{ 32 } =\frac{ 1 }{ 2 }\]

and

\[\frac{ 80 }{ 32 } = \frac{ 5 }{ 2 }\]

Answer 43

It felt like I did something wrong. =\

Answer 44

no you did right, you have the right solution, except you forgot the - sign at the 1/2

Answer 45

why does it feel wrong?

Answer 46

Because the number jumped from 80/32 to 5/2 , haha. Just I thought the number was too low, a bit of curiosity.

Answer 47

lol i forgot the - sign at the last one as well, we both did, that's quite funny.

Answer 48

oh well let me show you

Answer 49

look

\[80=40*2=20*2^2=10*2^3=5*2^4\]

Answer 50

\[32=16*2=8*2^2=4* 2^3=2^5\]

Answer 51

\[\frac{ 2^4 }{ 2^4 }=\frac{ 1 }{ 1 }=1\]

Answer 52

anything times 1 is still the same, so we can drop the 1

Answer 53

Interesting!! , I was not looking at it that way. But thats explainable!


The answer to the question would be: At points 1/2, 5/2, the ball will be 30 feet above the ground.

Answer 54

that's what i said, you can do this for ANY number that is not a prime number.

Answer 55

you basically take the number, and split it into the prime numbers it is made of, i always call it the number DNA.

Answer 56

you can most easily do it by starting with the lowest prime numbers but in some cases another prime number or even a multiple of several prime numbers may be easier.

Answer 57

Sweet!

Hey would you be down for just one more problem?

Answer 58

for example if a number ends with a 0 you can always divided by 2 and 5 (often multiple times)

Answer 59

sure.

Answer 60

A ball was thrown vertically upwards from top of a 25 foot tower with an initial velocity of 45 feet per second.

How high did the ball go?

Answer 61

ah, that's a physics formula

Answer 62

No wonder I purposely skipped it. I did everything else but that.

Answer 63

what's the gravity constant in feet? i only know the one for meters.

Answer 64

I have no clue. I am not in Psychics.

Answer 65

i will google

Answer 66

0.304800 let's just make it 0.3

Answer 67

doesn't matter that much for the purpose of explaining anyway

Answer 68

so, we have a few clues in the text.

Answer 69

first, the height at time 0 is 25.

Answer 70

Yepp and the the initial velocity is 45 feet per second.

Answer 71

beat me to it

Answer 72

now, it's not explicitly given but, we assume there is gravity there.

Answer 73

and since we defined gravity as 0.3 ft/s^2 (it's an acceleration, not a velocity)

Answer 74

and since the are also no other forces active (for the sake of simplicity we will ignore drag because drag will make your head spin)

Answer 75

we can make a formula for the height using these pieces of knowledge.

Answer 76

Thats a lot of new information. =\ , But go along. Im totally following you.

Answer 77

the +25 is pretty easy to explain, because the 25 at the start is not a function of time (otherwise it would be 0 and cancel out at time 0)

Answer 78

so, we write h=25 but obviously we are still missing some parts otherwise the ball would freeze in mid air an forever stay at 25 ft

Answer 79

so we need a function of time as well.

Answer 80

which is the speed initially given to it (45ft/s) so, every second of t, the ball moves 45ft

Answer 81

so after t, it moved 45, after 2t it moved 90 and so on

Answer 82

so what we have so far is \[h(t)=45t+25 (still~ incomplete!) \]

Answer 83

you understand it so far?

Answer 84

Lol. Im following let me soak it all in. Im re-reading everything. Standby.

Answer 85

alright

Answer 86

Alrighty. So how would t = 25?

Answer 87

in the function we have till now? it's not complete yet you know that right?

Answer 88

\[h(25)=45*(25)+25\]

Answer 89

which is 1150 btw

Answer 90

H = 46 ?

Answer 91

What do you mean by that, you want to know when H reaches 46?

Answer 92

Oh because h(25) = 25h

So in the equation its like:

25h = 1150

divide both sides by 25.

Answer 93

no you misinterpret that

Answer 94

Officially, when written like h(t)=something the h(t) means that h is a function of t

Answer 95

in this case, height is a function of time. (we want to know which height the ball has at any given time, that is the purpose of the function, to draw a graph of h against t)

Answer 96

so the h(25) in this case does not mean 25h but it simply means the value of t that h is a function of is 25.

Answer 97

in other words it sais, for every single t in the equation, replace it with 25. so you can ignore the numbers on the right.

Answer 98

Oh okay! I get it. Its just like f(x)