Question

A gardener wants the three rose bushes 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?

Answer 1

Okay from what this sounds like you will need to use a Cordinate plain

Answer 2

the idea is to find the center of the circle that passes through the three points. There is a shortcut fot this, but you haven't had it yet. You could construct them to find the center, but that would be an approximation. So here goes

A(1, 3), B(5, 11), and C(11, 4)
The slopeAB = 2, so the perp slope = -1/2, and the midpoint AB is (3, 7) so y = -.5(x - 3) + 7
The slope AC = 1/10 so perp slope = -10. mdpt AC = (6, 3.5) so y = -10(x - 6) + 3.5
Combine equations to get -.5(x - 3) + 7 = -10(x - 6) + 3.5
expand: -.5x + 1.5 + 7 = -10x + 60 + 3.5
Simplify: 9.5x = 55 x = 110/19 y = 213/38

So the center of the circle, found by determining where the perpendicular bisectors of two chords, is
(110/19, 213/38)
That is where the sprinkler should be placed


Now the distance to any of the bushes is found by the distance formula. Using the point (1, 3)

sqrt(((19 - 110)/19)² + ((114 - 213)/38)²) will be the distance from the sprinkler.

sqrt(22.94+ 6.787) = sqrt(29.73) = 5.45

so the equation describing the boundary is (x - 110/19)² + (y - 213/38)² = 29.73

Answer 3

thank you!

Answer 4

Anytime.