30 POINTSFencing encloses a rectangular backyard that measures 200 feet by 800 feet. A blueprint of the backyard is drawn on the coordinate plane so that the rectangle has vertices (0,0) (0,40) (10,40) and (10,0).A circular flower garden is dug to be exactly in the center of the backyard, with a radius of 60 feet. What is the equation of the flower garden represented on the blueprint? (x _ _)^2 + (y _ _)^2 = _​

Accepted Solution

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] β‡’ [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] β‡’[tex](x-5)^{2}+(y-20)^{2}=9[/tex]And that is the equation of the circle.