Question

An arrow is shot into the air from a building that is 32 ft. high. The height of the arrow after t seconds is given by the formula h(t)= -16t²+48t+32=0
At what times will the arrow be at a height of 64 ft?

Answer 1

Set the equation equal to 64 and solve.

Answer 2

first, use the derivative to find out the function for velocity

Answer 3

This is a superfluous step. The function gives height. You are solving for height. No need to differentiate. Just a quadratic function.

Answer 4

never mind, I thought too much

Answer 5

You are solving for time in a function of height...is what that should read. Apologies.

Answer 6

so i should still set the equation = to 64 and then what will i do?

Answer 7

Solve the equation for t.

Answer 8

so i should subtract the 64 or should i factor the formula and then set it = to 64 and solve?

Answer 9

You would have:
\[-16t^2 + 48t + 32 = 64\]
You can certainly factor a -16 out of the left

Answer 10

Then what do you have?

Answer 11

-16(t²-48t-32)=64?

Answer 12

No, 16 will also go into 48 and 32. This is not factored all of the way.

Answer 13

oops

Answer 14

-16(t²-3t-2)=64?

Answer 15

Yep, now divide out the -16.

Answer 16

=-4 and then i just factor it and then divide?

Answer 17

t²-3t-2=-4 what do i do now?

Answer 18

Yes, well you would factor this by completing the square. Then what do you have?

Answer 19

Hint: You can get REALLY close with (t - 1)(t - 2)

Answer 20

can i add the move the -2 from the left to the right to get =-4 +2 then i move that -2 to the left to get t²-3t+2?

Answer 21

Well, what happens when you distribute out (t-1)(t-2) ?

Answer 22

it equals t²-3t+2

Answer 23

Right, but that doesn't equate to our original left hand side. It's greater by 4. So how do we make the +2 into a -2 ?

Answer 24

to get the answer i have to complete the square right?

Answer 25

Yes and that's what we're doing. To change the last term (+2) into (-2) we have to subtract 4. So you have:
\[(t-1)(t-2) - 4 = -4\]
Now add the 4 to both sides, what do you have?

Answer 26

(t-1)(t-2)=0? i'm confused on how it went from t²-3t-2=-4 to get that

Answer 27

To complete the square we realize that we can get the first two terms easily, but the last term doesn't come out right:

(t-1)(t-2) = x^2 - 3x + 2

However, our original expression was x^2 - 3x - 2

Close but no cigar. We need to turn that +2 to a -2. To do this we subtract 4. Therefore we have the nice term: (t-1)(t-2)-4 = -4

Answer 28

We're just making the left hand side look different. Making it easier to work with.

Answer 29

where did you get the subtraction of -4 though is where i'm confused on

Answer 30

Because when we foil out the expression (t-1)(t-2) and subtract 4, we get our ORIGINAL expression back.

Answer 31

Ok, FOIL out (t-1)(t-2)

Answer 32

so (t-1)(t-2) is = to t²-3t+2 and what you want me to do is factor out of t²-3t-2, (t-1)(t-2)?

Answer 33

Well, follow me here. If (t-1)(t-2) = t^2 - 3t + 2, what happens when we subtract 4?

Answer 34

if we subtract 4 it turns into t^2-3t-2

Answer 35

Thats our ORIGINAL expression isnt it? :]

Answer 36

yes it is

Answer 37

Right, if we didnt subtract out the 4, then we have just borked the entire equation and the solutions would be incorrect. The key in this is making it LOOK different while still meaning the same thing. We are ignoring the right hand side, btw...

Answer 38

If we didn't complete the square we would have to use the quadratic formula which is cumbersome.

Answer 39

\[x^2 - 3x - 2 = (x-1)(x-2)-4\]

Answer 40

oh okay i see now

Answer 41

Right, so now add the 4 and you simply have (t-1)(t-2) = 0

Answer 42

Now what are the solutions?

Answer 43

t=1 or t=2

Answer 44

t = 1 AND t = 2.
At 1 second, it's at 64ft on the way UP, at 2 seconds its at 64ft on the way DOWN.

Answer 45

OH i see now and i fully get it. thank you so much

Answer 46

No problem.