Question

Catapults are being used to hit an island. At 20 kilometers from shore the incoming boulders were located at 2800 meters of altitude. At 40 kilometers from shore, they were spotted at 4200 meters of altitude. And at 70 kilometers out, the boulders were passing at 5550 meters of altitude.

How far from their shore is their enemies catapult?

Are their shots too short or too long to hit us and by how much?

Answer 1

We need to use the equation of a parabola here? do you know it?

Answer 2

y = ax² + bx + c

Answer 3

I believe thats it

Answer 4

correct

Answer 5

now lets use that to solve the problem

Answer 6

let x and y be in kilimoters

Answer 7

we have three points to work with

Answer 8

(20,2.8) (40,4.2) (70,5.5)

Answer 9

We must use these point to solve: y = ax² + bx + c

Answer 10

are you paying attention or not?

Answer 11

yes

Answer 12

now i will set up the equations, and you must solve them or at least try, ok?

Answer 13

ok

Answer 14

c + 20b + 400a = 2.8,
c + 40b + 1600a = 4.2, and
c + 70b + 4600a = 5.5.

Answer 15

which am I solving for a, b, or c

Answer 16

now you understand how i got these right, i places the x coordinated from our 3 points ((20,2.8) (40,4.2) (70,5.5)) that means the 20 the 40 and 70 replaced the x's in the equation y = ax² + bx + c

Answer 17

well, we are going to solve these like we would solve a system of equations

Answer 18

yes I get stuck right here where we are now

Answer 19

now subtract the 1st equation from the second please

Answer 20

and then tell me what you get

Answer 21

ok

Answer 22

20b+1200a=1.4

Answer 23

okay, good now subtract the 1st equation from the third, and tell me what you get

Answer 24

50b+4200a=2.7

Answer 25

correct

Answer 26

now subtract 5/2 of the 1st equation from the second

Answer 27

I dont get it what do u mean subtract 5/2

Answer 28

Ok I subtract 5/2 from 1200a correct but when I do it. It doesn't look right

Answer 29

Multiply 1200, by 2.5 and subtract it from 4200

Answer 30

ok

Answer 31

The multiply 20 by 2.5 and subtract it from 50

Answer 32

Then multiply 1.4 by 2.5 and subtract it from 2.7

Answer 33

did you get it?

Answer 34

I got -2950

Answer 35

you should have gotten 1200a=-0.8

Answer 36

Check this out

Answer 37

ok hold on trying it again

Answer 38

ok 1200*2.5=3000

4200-3000=1200

Answer 39

(2.5)(20b+1200a=1.4)=50b+3000a=3.5

Answer 40

now subtract that from our second equation: 50b+4200a=2.7

Answer 41

we get:
50b+4200a=2.7
- 50b+3000a=3.5
----------------
=

Answer 42

1200a=-0.8

Answer 43

correct

Answer 44

now lets solve for a

Answer 45

how can we do that?

Answer 46

thats right, we divide both sides of 1200a=-0.8, by 1200 and get?

Answer 47

a=-0.00066667

Answer 48

oh ok if we divided both sides by 1200 we get a=-0.8?

Answer 49

i mean a=-8/12,000

Answer 50

ok

Answer 51

not* you see how i changed -0.8 to -8 and 1200 to 12000

Answer 52

note*

Answer 53

yes

Answer 54

okay, now we have found a, lets find c and b. To find b, simply replace a with -8/12000 in the equation 50b + 4200a = 2.7

Answer 55

and solve, tell me what you get

Answer 56

0.11

Answer 57

b=0.11

Answer 58

correct

Answer 59

now to find c, simple replace a with -8/12000 and b with 0.11(or 5.5/50) in the equation c + 20b + 400a = 2.8

Answer 60

ok one sec

Answer 61

c=13/15

Answer 62

hmm

Answer 63

I did as u said c+20(0.11)+400(-8/12000)=2.8

Answer 64

correct

Answer 65

now we have all three values for a,b and c

Answer 66

ok

Answer 67

now to determine where the boulders will hit solve ax² + bx + c = 0.

Answer 68

ok

Answer 69

This will give give the x values for where it was launched from and where it will hit.
Using these values and the fact that the place you are at is at (0,0),
the distance from you can be found.

Answer 70

now:
If the x value is positive, the boulders fell short.
If the x value is negative, the boulder went too far

Answer 71

x=172.5347,-7.534716

Answer 72

is that correct?

Answer 73

very good

Answer 74

Thank you very very much

Answer 75

yw

Answer 76

Are u still here I have one more question

Answer 77

i hope its not as long as this one :)

Answer 78

lol no but if the questions ask me How far from their shore is their enemies catapult?

Are their shots too short or too long to hit us and by how much? How would
I plug in my answers here?

Answer 79

I hate word problems they confuse me

Answer 80

right we answered this already right?

Answer 81

Yes just not sure how to respond if x=172.5347,-7.534716

Answer 82

if x is positive there shots fell short, if x is negative there shots went to far

Answer 83

ok

Answer 84

ok got it thanks again