Question

Can someone make sure I'm doing this right?:
An archer releases an arrow from a shoulder height of 1.39 m. When
the arrow hits the target 18 m away, it hits point A. When the target is
removed, the arrow lands 45 m away. Find the maximum height of the
arrow along its parabolic path. The top of the target is 1.5 meters from the ground.

I used the points (0,1.39) (18,1.26) (45,0)
Where c=1.39
I then got the equations:
-.13=324a+18b and -1.39=2025a+45b and solved for a and b
a=-.0008765
b=.008555
I then used x=-b/2a and got x=4.89

Answer 1

This is where I stopped because I wasn't sure what else to do

Answer 2

the initial angle is zero?

Answer 3

What is the initial angle?

Answer 4

thats not given

Answer 5

but you are given some points on the parabolic trajectory

Answer 6

Hmm I don't know I think your on the right track though, just not sure where to go from here

Answer 7

you have points
(0, 1.39) (45,0)

Answer 8

Yeah and I used them.

Answer 9

and you have (18, 1.5 - 16/100 )

Answer 10

woops, should be 24 cm

Answer 11

( 18, 1.26 )

Answer 12

ok you can use TI 84 and curve fit your points

Answer 13

I don't actually have a graphing calculator

Answer 14

y = ax^2 + bx + c
a = -8.7654
b = .00856
c = 1.39

Answer 15

Is that all there is? Or do I have to do more?

Answer 16

what you have a system of equations

Answer 17

you have points {(x,y) : (0,1.39), (18,1.26), (45,0)}
plug them into ax^2 + bx + c = y

a(0)^2 + b(0) + c = 1.39
a(18)^2 + b(18) + c = 1.26
a(45)^2 + b(45) + c = 0

Answer 18

so the first equation tells us that c = 1.39

Answer 19

plug that c value into equation 2 and equation 3

a(18)^2 + b(18) + 1.39 = 1.26
a(45)^2 + b(45) + 1.39 = 0

Answer 20

324a + 18b = 1.26 - 1.39
2025a + 45b = -1.39

Answer 21

324a + 18b = -0.13
2025a + 45b = -1.39

Answer 22

can you solve that system of equations?

Answer 23

Yup just a second

Answer 24

a=-.0008765

Answer 25

b=.008555

Answer 26

@perl

Answer 27

Still haven't figured this out

Answer 28

your arithmetic is wrong somewhere

Answer 29

Well I'm not sure what I'm doing wrong then because I keep getting the same answer

Answer 30

we want to solve
324a + 18b = -0.13
2025a + 45b = -1.39

agreed?

Answer 31

Yup

Answer 32

how would you like to solve it, by substitution or elimination?

Answer 33

I used elimination

Answer 34

ok

Answer 35

elimination is a bit of work

Answer 36

I'm getting

a=-0.0008765
b=0.008555
c = 1.39

just as you are. Geogebra confirms it

Answer 37

So then thats right?

Answer 38

you used the points
(0,1.39) (18,1.26) (45,0) ?

Answer 39

Yeah

Answer 40

oh , i left out x 10 ^-4

Answer 41

y = ax^2 + bx + c
a = -8.7654 * 10^-4
b = .00856
c = 1.39

Answer 42

So what would be my next step? I'd have to use x=-b/2a right

Answer 43

correct

Answer 44

the calculation for x = -b/(2a) is off by a digit though

x = -b/(2a)
x = -(0.008555/(2*(-0.0008765))
x = 4.880205
x = 4.88 if you round to 2 decimal places

Answer 45

x = -.008556 / ( 2 * -.0008765432)

Answer 46

Oh okay. This is the part I got stuck at, how do I find y?

Answer 47

once you have the x coordinate of the vertex, you plug it into the function that represents the arc to get the y coordinate of the vertex

Answer 48

so plug it into f(x)

Answer 49

Just f(x)?
So f(4.88)?

Answer 50

what is f(4.88) equal to?

Answer 51

remember that f(x) is the quadratic that goes through all 3 points

Answer 52

I don't know how to find that

Answer 53

what is f(x) ?

Answer 54

what is the actual function or quadratic

Answer 55

you found the values of a,b,c

they will be used in f(x) = ax^2 + bx + c

Answer 56

So then f(4.88)=-.008765x^2+.008555+1.39

Answer 57

close

Answer 58

you forgot about the x attached to b

Answer 59

oh oops

Answer 60

and you would replace EVERY x with the 4.88

Answer 61

Okay

Answer 62

Is it 1.22302?

Answer 63

it's a little short

Answer 64

I'm getting a slightly larger answer.

Answer 65

Wait I think I did that wrong. 1.33577947?

Answer 66

you should have

-0.0008765(4.88)^2+0.008555(4.88)+1.39 = 1.41

you had 2 zeros in the first coefficient when it should have been 3

Answer 67

btw I remember doing this a while back with another person and I made up this drawing

Answer 68

so that gives a visual summary in a way

Answer 69

Okay so I got 1.408767606?

Answer 70

Oh well that makes it a bit easier lol thank you

Answer 71

that's closer, so if you round what you got, you would have 1.41

Answer 72

So then the maximum height is (4.88,1.41)

Answer 73

that is the point where the max height happens

the actual max height is 1.41 meters. This is the y coordinate of the vertex point

Answer 74

the 4.88 is the x coordinate

that tells you how far away horizontally the arrow is from the archer

Answer 75

Oh alright

Answer 76

And thats all then?

Answer 77

yes

Answer 78

Thank you so much for sticking around and walking me through it!!

Answer 79

sure thing