You know that the length and the width of the rectangular backyards are:
[tex]\begin{gathered} l=600ft \\ w=300ft \end{gathered}[/tex]
Knowing the coordinates of the vertices of the rectangle, you can plot them on a Coordinate Plane and draw the rectangle. Notice that the vertices are:
[tex]\mleft(0,0\mright),\mleft(0,60\mright),\mleft(30,60\mright),\mleft(30,0\mright)[/tex]
See the picture below:
According to the information given in the exercise, a circular flower garden is dug to be exactly in the center of the backyard. The radius of this circle is:
[tex]r=60ft[/tex]
If you draw the diagonals of the rectangle, you can find its center and therefore, the center of the circle. See the picture below:
Now you know the center of the circle.
By definition, the equation of a circle is:
[tex]\mleft(x-h\mright)^2+\mleft(y-k\mright)^2=r^2[/tex]
Where "r" is the radius of the circle and its center is:
[tex](h,k)[/tex]
Since you already know the center of the circular flower garden and its radius, you only need to substitute values into the equation and simplify:
[tex]\begin{gathered} (x-15)^2+(y-30)^2=6^2 \\ (x-15)^2+(y-30)^2=36 \end{gathered}[/tex]
Therefore, the answer is:
[tex](x-15)^2+(y-30)^2=36[/tex]
