Question

A gardener wants the three rosebushes in her garden to be watered by a rotating water sprinkler. The gardener draws a diagram of the garden using a grid in which each unit represents 1 ft. The rosebushes are at (1, 3), (5, 11), and (11, 4). She wants to position the sprinkler at a point equidistant from each rosebush. Where should the gardener place the sprinkler? What equation describes the boundary of the circular region that the sprinkler will cover?

P.S: I saw some answers from other websites but the explanations are quite vague-- I prefer explanations over answers (It's okay if you'll just going to list down the formula I can use in a corresponding manner. Man, I just want to sleep. Thank u in advance!)

Answer 1

Hello! (for future references please refrain from using inappropriate words)

Lets go through this problem step by step

Answer 2

Thank you! really sorry, was just really in a bad mood since I've tried understanding it but nothing really comes to mind.

Answer 3

It's all good (:

Answer 4

First step - We set up the problem

\[(1 - h)^2 + (3 - k)^2 = (5 - h)^2 + (11 - k)^2\]

Answer 5

There are various of ways to solve it, but I'm not sure what method I should use though.

Answer 6

Ohh okay2

Answer 7

Okay so what is the next step?

Answer 8

We seperate into parts and our outcome should be-
\[1 - 2h + 9 - 6k = 25 - 10h + 121 - 22k\]

Answer 9

Oh! So you just need to expand the binomial? (like make it into a trinomial)

Answer 10

The next step is to combine like terms right?

Answer 11

Yes

Answer 12

\[10 - 2h - 6k = 146 - 10h - 22k\]

Answer 13

Then we divide both sides of the equation by 2 -

\[10 - 2h - 6k = 146 - 10h - 22k\]

Answer 14

ooh, i have another question.

Answer 15

We're still not done here. ;O

Answer 16

oh sorry hehe, ill ask after.

Answer 17

Lol, Do you know what the next step is?

Answer 18

\[5h - h + 11k - 3k = 73 - 5\]
\[4h + 8k = 68\]
\[h + 2k = 17\]

Answer 19

\[h = 17 - 2k\]

Answer 20

But there's even more to this problem-

Answer 21

oh no--

Answer 22

Lol, but it's alright we can do this
\[h = 17 - 2k\]
^ What do we get now?

Answer 23

the value of h?

Answer 24

\[h = (17 * 19 - 213) / 19\]

Answer 25

Now what is 17*19

Answer 26

323

Answer 27

then we subtract it with 213

Answer 28

Alright now we plug that in -
\[h = (323 - 213) / 19\]

Answer 29

Leaves you with?

Answer 30

110/19

Answer 31

or 5.8

Answer 32

Yes Correct, and now-

Answer 33

For (1,3) (11,4)

Answer 34

\[(1 - h)^2 + (3 - k)^2 = (11 - h)^2 + (4 - k)^2\]

Answer 35

Now what do you do next?

Answer 36

just like we did before, we expand the binomial and if im correct i think it'd look like this-
1 - 2h + 9 - 6k = 121 - 22h + 16 - 8k

Answer 37

Yes exactly

Answer 38

And then- \[10 - 2h - 6k = 137 - 22h - 8k\]

Answer 39

then, combine like terms: 20h + 2k = 127

Answer 40

Yes correct

Answer 41

\[20 * (17 - 2k) + 2k = 127\]

Answer 42

340 - 40k + 2k = 127

Answer 43

\[340 - 40k + 2k = 127\]
\[-38k + 340 - 127 = 0\]
\[213 = 38k\]
\[k = 213/38\]
\[Y = 5.6\]

Answer 44

Okay, so you're answer is? :D

Answer 45

Your*

Answer 46

h= 5.8
k= 5.6

Answer 47

sooo, that's the coordinates of the center

Answer 48

The boundary equation is-

(x - 110/19)² + (y - 213/38)² = Square root 29.73

or

(x -5.8)² + (y -5.6)² =Square root 29.73.

Answer 49

and to get the boundary smth, we just substitute the value of h and k to the equation of the circle formula

Answer 50

OOOhhh

Answer 51

Lol, Good Job!!

Answer 52

Now we're done here (:

Answer 53

Thank you sm for teaching me!! :)))

Answer 54

Feel free to leave a medal XD

Answer 55

wahahhaha idk how, but thanks! ive just recently join. You earned yourself another fan!

Answer 56

Lol It's all good, the medals are right over here

Answer 57

And thanks for the fan!

Answer 58

hi again, uhh u awake? Forgot I have to ask a question. How come when we solved for the binomial (to make it into trinomial) (1−h)^2+(3−k)^2=(5−h)^2+(11−k)^2, we didn't get a variable with an exponent after?

Answer 59

like shouldnt we get h^2 when we solved for (1-h)^2? Thanks in advance!