Answer:
The equation is [tex](x-5)^{2}+(y-20)^{2}=9[/tex]
Step-by-step explanation:
Given the following equation in the plane :
[tex](x-a)^{2}+(y-b)^{2}=R^{2}[/tex]
The set of points [tex](x,y)[/tex] that satisfy the equation graph a circle in the plane.The circle is centered at [tex](a,b)[/tex] and the radius of the circle is R.
For example, the set of points that satisfy : [tex](x-2)^{2}+(y-3)^{2}=9[/tex]
graph a circle of radius [tex]\sqrt{9}=3[/tex] centered at the point [tex](2,3)[/tex]
The first step to solve this exercise is to find the center of the circle.
The rectangle has vertices [tex](0,0),(0,40),(10,40)[/tex] and [tex](10,0)[/tex] so they will form a rectangle with width 10 units and height 40 units.
Given this situation, the center of the rectangle is at [tex](5,20)[/tex] (Half of the width in the first coordinate and half of the height in the second one)
The equation of the flower garden will be
[tex](x-5)^{2}+(y-20)^{2}=R^{2}[/tex]
The final step is to find the value of R.
Given that the rectangular backyard width is 200 feet and this is represented with a rectangle in the blueprint with width 10 units
[tex]10units=200feet[/tex] ⇒ [tex]60feet=\frac{(60).(10)}{200}units=3units[/tex]
The radius of 60 feet is represented with 3 units in the blueprint.
Now we replace [tex]R=3[/tex] in the equation :
[tex](x-5)^{2}+(y-20)^{2}=3^{2}[/tex] ⇒
[tex](x-5)^{2}+(y-20)^{2}=9[/tex]
And that is the equation of the circle.
