Question

An archer releases an arrow from a shoulder height of 1.39m. When the arrow hits the target 18m away, it hits point A (at 1.28m). When the target is removed, the arrow lands 45m away. Find the maximum height of the arrow along its parabolic path.

Can someone PLEASE give a detailed description of how to solve this and show work? I've been sitting here for over an hour trying to figure this out.

Answer 1

parabolic path means a parabola, which means an equation of the form
y = ax^2 + b x + c

here y will be the height, x is the distance from the archer. the max height will be the vertex of the parabola, which occurs at x= -b/(2a)

so you have to find the coefficients a, b and c.

Answer 2

An archer releases an arrow from a shoulder height of 1.39m
we put the archer at x=0, so this point is (0,1.39)

When the target is removed, the arrow lands 45m away.
the height of the arrow on the ground is 0, so this point is (45,0)

we need one more point.

Answer 3

he arrow hits the target 18m away, it hits point A (at 1.28m).

x= 18 y will be the height above the ground. But I don't know the height of point A. Is there a picture ?

Answer 4

Wouldn't 1.28m be the height?

Answer 5

maybe... it is not clear from the write-up. Is there a picture ?

Answer 6

Yeah hold on!

Answer 7

had this question the other day :)

Answer 8

you are given 3 points to define a parabola with

Answer 9

(0,shoulder)

(18,A)

(45,0)

Answer 10

yes, the height is 1.28
so the third point is (18.1.28)

Answer 11

use each point in the "generic equation"
y = a x^2 + b x + c
to generate 3 equations with 3 unknowns.

can you do that ?

Answer 12

well, we would already now one of the "unknowns" :) reducing it to 2 eqs in 2 unknowns

Answer 13

I'm still really confused! How do I write the equations with the unkowns and what do I do with them?

Answer 14

to start at the beginning:

we are told we have a parabola (parabolic path)
we (might? should?) know a parabola is
y = a x^2 + b x + c where a , b and c are unknown coefficients

to solve for 3 unknowns, we need 3 different equations.
notice we have 3 (x,y) points. If we take an (x,y) point example: (0,1.39)
and replace x and y with 0 and 1.39, we will get an equation with unknowns a,b and c

can you do that ?

Answer 15

So it would be like a system of equations thing?

Answer 16

yes

Answer 17

Okay hold on let me see if I can do this..

Answer 18

Okay I'm really bad at 3 part systems of equations...
I got these equations:
1.39=a(0)^2 + b(0) +c
0=a(45)^2 + b(45) + c
1.28 = a(18)2 + b(18) + c

So then how do I find the values of the variables? Having 3 equations always messes me up!

Answer 19

you can "solve" the first one easily.

Answer 20

Oh yeah so wouldn't it be c=1.39?

Answer 21

yes. so replace c with 1.39 in the other 2 equations. now it's 2 eq's and 2 unknowns

Answer 22

So then would I do like the elimination method?
Sorry I'm usually able to figure out these problems but today my mind has just gone blank...

Answer 23

these are ugly numbers.

but you now have
0=a(45)^2 + b(45) + 1.39
1.28 = a(18)2 + b(18) + 1.39

if we add -1.28 to both sides of the 2nd equation we get

a(18)^2 + b(18) + 0.11=0


you now have
\[ a(18)^2 + b(18) + 0.11=0 \\a(45)^2 + b(45) + 1.39=0
\]

Answer 24

I would divide the first equation by 18, and the 2nd by 45 to get
\[ 18a + b +\frac{0.11}{18} = 0 \\ 45a + b + \frac{1.39}{45} = 0\]

Answer 25

Okay cool...what's the next step?

Answer 26

Do we eliminate b?

Answer 27

subtract the two equations

Answer 28

Okay I got -27a - 0.0247 = 0?

Answer 29

keep lots of decimals for this problem.... but now solve for a

Answer 30

a = -9.15?

Answer 31

a= 0.02477777/-27

that can be -9.15

Answer 32

*can't

Answer 33

Okay I got -9.17695185E-4
What does the E mean?

Answer 34

scientific notation. E stands for "exponent"
that number is
\[ -9.17695185\cdot 10^{-4} = -0.000917695185\]

but it is easier to keep it in scientific notation

Answer 35

now solve for b using
\[ 18a + b +\frac{0.11}{18} = 0 \]

Answer 36

My calculator says "error" when I try to multiply 18 and -9.17695185 x 10^-4, so should I just use the decimal number instead??

Answer 37

does your calculator have an "EXP" button ? if so, try 2 EXP 2 =
you should see 200

Answer 38

Yeah I found it!

Answer 39

so type in -9.17695185 EXP -4
(and save it in memory if you know how)

Answer 40

Okay I got b = -0.0104074023

Answer 41

that looks good, except it should be positive

Answer 42

Oh right sorry I didn't mean to put that negative there! Okay so now what do we do??

Answer 43

Find the maximum height of the arrow along its parabolic path.

that means find the vertex of the parabola
see http://hotmath.com/hotmath_help/topics/vertex-of-a-parabola.html

Answer 44

Okay so the vertex is (5.67, 1.42)

Answer 45

So the max. height is 1.42m.
Thank you SO much for your help @phi I couldn't have done it without you! <3

Answer 46

yes, that looks good