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 rose are at (1,3), (5,11), and (11,4). She wants to position the sprnkler at the a point equidistant from each rosebush. Where should the grdener place the sprinkler? What equation describes the boundary of the circular region that the sprinkler will cover?

Answer 1

To find the point equidistant from each rosebush, you can calculate the average x-coordinate and the average y-coordinate of the three rosebushes. The average x-coordinate is (1 + 5 + 11) / 3 = 17 / 3 ≈ 5.67, and the average y-coordinate is (3 + 11 + 4) / 3 = 18 / 3 = 6.

So, the gardener should place the sprinkler at the point (5.67, 6) to ensure it's equidistant from each rosebush.

The equation that describes the boundary of the circular region that the sprinkler will cover is:
(x - 5.67)^2 + (y - 6)^2 = r^2
where (5.67, 6) is the coordinates of the center of the circle, and r is the radius of the circular region.

Answer 2

let O (x,y) be position of sprinkler.
and A(1,3),B(5,11),C(11,4)
OA=OB=OC
\[(x-1)^2+(y-3)^2=(x-5)^2+(y-11)^2=(x-11)^2+(y-4)^2\]
taking first two
\[x^2-2x+1+y^2-6y+9=x^2-10x+25+y^2-22y+121\]
\[-2x-6y+10=-10x-22y+146\]
\[-2x+10x-6y+22y=146-10\]
8x+16y=136
divide by 8
x+2y=17 ...(1)
taking last two
\[x^2-10x+25+y^2-22y+121=x^2-22x+121+y^2-8y+16\]
\[-10x-22y+146=-22X-8y+137\]
\[-10x+22x-22y+8y=137-146\]
12x-14y=-9 ...(2)
multiply (1)by 7 and add in (2)
12x+7x-14y+14y=-9+119
19x=110
\[x=\frac{ 110 }{ 19}\]
from (1)
\[\frac{ 110 }{ 19 }+2y=17\]
\[2y=17-\frac{ 110 }{ 19 }=\frac{ 323-110 }{ 19 }=\frac{ 213 }{ 19}\]
\[y=\frac{ 213 }{ 38}\]
\[r=distance~\between~(1,3)~and ~(\frac{ 110 }{ 19 },\frac{ 213 }{ 38})\]
\[r=\sqrt{\left( \frac{ 110 }{ 19 }-1 \right)^2+\left( \frac{ 213 }{ 38}-3 \right)^2}\]
\[r=\sqrt{\left( \frac{ 110-19 }{ 19 } \right)^2+\left( \frac{ 213-114 }{ 38 } \right)^2}\]
\[r=\sqrt{\left( \frac{ 91 }{ 19 } \right)^2+\left( \frac{ 99 }{ 38} \right)^2}\]
\[r=\frac{ 1 }{ 38 }\sqrt{4\times8281+9801}\]
\[r=\frac{ 1 }{ 38 }\sqrt{42925}=\frac{ 5 }{ 38 }\sqrt{1717}\]
or
\[r=\sqrt{\frac{ 42925 }{ 1444}}\]
so eq. of circle is
\[\left( x-1 \right)^2+\left( y-3\right)^2=\frac{ 42925 }{ 1444}\]

Answer 3

To solve this problem, we need to find the point that is equidistant from the three rosebushes and then determine the equation of the circular region that the sprinkler will cover.

Given information:
- The rosebushes are located at (1,3), (5,11), and (11,4).
- The gardener wants to position the sprinkler at a point equidistant from each rosebush.

Step 1: Find the point that is equidistant from the three rosebushes.
Let the coordinates of the sprinkler be (x, y).
The distances from the sprinkler to the three rosebushes are:
- Distance to (1,3): √[(1-x)^2 + (3-y)^2]
- Distance to (5,11): √[(5-x)^2 + (11-y)^2]
- Distance to (11,4): √[(11-x)^2 + (4-y)^2]

Since the gardener wants the sprinkler to be equidistant from the three rosebushes, we can set the three distances equal to each other:
√[(1-x)^2 + (3-y)^2] = √[(5-x)^2 + (11-y)^2] = √[(11-x)^2 + (4-y)^2]

Solving this system of equations, we get the coordinates of the sprinkler as (6, 6).

Step 2: Determine the equation of the circular region that the sprinkler will cover.
The radius of the circular region is the distance from the sprinkler to any of the rosebushes.
Radius = √[(1-6)^2 + (3-6)^2] = √25 = 5 ft

The equation of the circular region is:
(x - 6)^2 + (y - 6)^2 = 25

Therefore, the gardener should place the sprinkler at the point (6, 6), and the equation of the circular region that the sprinkler will cover is (x - 6)^2 + (y - 6)^2 = 25.

Answer 4

that sure is a lot of math

Answer 5

🤫🧏