Question

Can someone check my answer???
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 represent 1 ft. The rosebushes are at (1,3),(5,11), and (11,4). she wants to position the sprinkler at a point equidistance from each rosebush. where should the gardener place the sprinkler? what equation describes the boundary of the circular region that the sprinkler will cover?

Answer 1

There are some assumptions, firstly that the sprinkler only reaches the three bushes and goes no further, but anyways under that assumption you need to find the center of the circle that contains the three points.

Answer 2

(x-h)^2+(y-k)^2=r^2 is the general circular formula. There may be a simpler method but I think you will need to solve the system of three equations when you plug each point into the circle equation and solve for h,k, and r.

Answer 3

I think you and @ellamoyseyuk are doing the same course

Answer 4

1. They all intersect at (5.8, 5.6).

(1,3),(5,11)

(1 - h)^2 + (3 - k)^2 = (5 - h)^2 + (11 - k)^2
1 - 2h + 9 - 6k = 25 - 10h + 121 - 22k
10 - 2h - 6k = 146 - 10h - 22k
5 - h - 3k = 73 - 5h - 11k
5h - h + 11k - 3k = 73 - 5
4h + 8k = 68
h + 2k = 17
h = 17 - 2k

h = 17 - 213/19
h = (17 * 19 - 213) / 19
h = (323 - 213) / 19
h = 110/19
X = 5.8

(1,3)(11,4)

(1 - h)^2 + (3 - k)^2 = (11 - h)^2 + (4 - k)^2
1 - 2h + 9 - 6k = 121 - 22h + 16 - 8k
10 - 2h - 6k = 137 - 22h - 8k
20h + 2k = 127
20 * (17 - 2k) + 2k = 127
340 - 40k + 2k = 127
-38k + 340 - 127 = 0
213 = 38k
k = 213/38
Y = 5.6

This is the boundary equation (x - 110/19)² + (y - 213/38)² = Sqrt 29.73 or (x -5.8)² + (y -5.6)² =sqrt 29.73.

Answer 5

is that the correct answer?

Answer 6

@phi can u check my answer?

Answer 7

ok, except it should be r^2 (not r) in the equation
in other words, it should be 29.73 not sqr(29.73)

also, it's more accurate to keep the exact value for r^2 which I think is
42925/1444

Answer 8

1. They all intersect at (5.8, 5.6).

(1,3),(5,11)

(1 - h)^2 + (3 - k)^2 = (5 - h)^2 + (11 - k)^2
1 - 2h + 9 - 6k = 25 - 10h + 121 - 22k
10 - 2h - 6k = 146 - 10h - 22k
5 - h - 3k = 73 - 5h - 11k
5h - h + 11k - 3k = 73 - 5
4h + 8k = 68
h + 2k = 17
h = 17 - 2k

h = 17 - 213/19
h = (17 * 19 - 213) / 19
h = (323 - 213) / 19
h = 110/19
X = 5.8

(1,3)(11,4)

(1 - h)^2 + (3 - k)^2 = (11 - h)^2 + (4 - k)^2
1 - 2h + 9 - 6k = 121 - 22h + 16 - 8k
10 - 2h - 6k = 137 - 22h - 8k
20h + 2k = 127
20 * (17 - 2k) + 2k = 127
340 - 40k + 2k = 127
-38k + 340 - 127 = 0
213 = 38k
k = 213/38
Y = 5.6

This is the boundary equation (x - 110/19)² + (y - 213/38)² = 42925/1444.

Answer 9

is that more correct now?

Answer 10

yes. you can check by putting in one of the points, say (1,3)
pasting the following into google (or use a calculator)
(1 - 110/19)^2 + (3 - 213/38)^2 =