Question

Dealing with Calculus rates of change: If a ball is thrown vertically upward with a velocity of 80ft./second, then its height after t seconds is s=80t-16t^2.
a) what is the maximum height reached by the ball?
b) what is the velocity of the ball when it is 96ft. abov the ground on its way up? On its way down?

Answer 1

a) Let s' = 0 --> t => max s

Answer 2

so i set s' equal to zero, then solve for t, correct?

Answer 3

Yes, s' = 0 --> t!

Answer 4

ok, I got 5ft./2 seconds. For the second part ,would I substitute 96 in for s'?

Answer 5

So max height is:
s ( 2.5 ) = ....

Answer 6

ok. Why do you have to set s' to zero?

Answer 7

I'm really bad with math. Trying to understand this. Ok, so the answer to a) is 2.5 ft./s. How do I tackle question b)?

Answer 8

a) Let s' = 0 --> t => max s
You haven't finished , t = 2.5 sec.
Now plug in S ( 2.5) to find max height!

Answer 9

ok, did that and got 100 ft./s

Answer 10

40ft/s

Answer 11

Ok, i took the deriv. of s=80t-16t^2 to get s'=80-32t.

set s'=0 to get 0=80-32t, which gave me t=2.5.
I plugged my value of t back into my original equation to get 100ft./s : s(2.5)= 80(2.5)-16(2.5)^2= 100ft./s
I also plugged it into the s' equation: 80-32(2.5)= 40ft./s

Answer 12

This is how you show your calculation, not just plug the numbers in:
s(2.5)= 16 ( 5*2.5 - 2.5²)
= 16 ( 12.5 - 6.25 )
= 16 * 7.25 = ....?

Answer 13

you used power rule

Answer 14

Nothing fancy, just plug the numbers in! In order to simplify the calculation, I factorize 16 out. That's it!

Answer 15

OH!!!!!

Answer 16

won't the answer still be 100 ft./s? because 2.5^2 is 6.25, and 12.5-6.25 = 6.25. Then
16*6.25= 100

Answer 17

Did I miss something?

Answer 18

Oops, I misread my own handwriting!
Yes, 16 * 6.25 = 100 ( Sorry!)

Answer 19

Its ok dude... or dudette! At least I'm paying attention, right? lol

Answer 20

So, a) Max h = 100 ft

Answer 21

YAY!!!

Answer 22

Pay attention to the units t = 2.5 secs, Max height = 100 fts

Answer 23

i just realized that my answer didn't make sense adding the ft./s on there. You're right, its looking for height

Answer 24

that the ball is 96 ft. in the air and looking for the velocity while its up in the air and when its falling back down.

Answer 25

I mean where do you plug 96 in?

Answer 26

and velocity is speed with direction, and we've got direction, now we need speed. Solving for s?

Answer 27

Plug in 96ft where s is?

Answer 28

Yes!

Answer 29

YAY!!!

Answer 30

DO i plug 96 into the original equation or the s' equation?

Answer 31

Plug into S, not S' ( because S is height, S' is velocity )

Answer 32

I MUST remember that fact. Ok, let me try this

Answer 33

damn t variables, throwing me off

Answer 34

S = 96 --> t => V :)

Answer 35

I got -139,776

Answer 36

I did something wrong. s(96)= 80(96)-16(96)^2

Answer 37

No, Plug 96 into where S is ( not where t is):
S = 16 ( 5t - t ²)
96 = 16 ( 5t - t ²)

Answer 38

Can you find t

Answer 39

that's where I'm stuck

Answer 40

96 = 16 ( 5t - t ²)
96/16 = 5t - t ²
Can you continue?

Answer 41

\[-t=\sqrt{6-5t}\]

Answer 42

I assume that t needs to be positive, AND not underneath the radical

Answer 43

96/16 = 5t - t ²
t ² -5t + 6 = 0

Answer 44

OH!!!!!

Answer 45

I'm a big stupid :(

Answer 46

ok, so now t= (2,3)

Answer 47

now I plug both of my answers back into the original equation and solve for each respectively?

Answer 48

NO, plug t into S' => V

Answer 49

ok, and I got.... 16 ft. and -106 ft.

Answer 50

ft./s

Answer 51

Can you show your calculation as I showed you?

Answer 52

s'= 80-32(2)= 80-64=26 ft./s
s'= 80-32(2)= 80-96=-106 ft./s

Answer 53

whoops, its 1, NOT 26

Answer 54

oomg, 16

Answer 55

V (3) = ...?

Answer 56

V(3)= s'= 80-32(3)= 80- 96=-16

Answer 57

-16 ft.s

Answer 58

Yup, now we arrive the same result this time =)

Answer 59

V(3) = -16 ft/sec

Answer 60

YAY!!!! danke shon Chlorophyll!!! you are very patient! I GREATLY appreciate your help!!!!

Answer 61

gracias, thank you!!! Whichever language you like hearing "thank you" in

Answer 62

well, at least anything in spanish, german, or english

Answer 63

i took half that time, for one question. I know I'm difficult, but I don't try to be

Answer 64

I did, thank you very, very much. I've got to go study anatomy. I hope that you'll be around to help again, soon. You have a good night/day, depending where you're located

Answer 65

Good luck with your quizzes, test, exams, .... :)