Question

30 POINTS

Answer 1

Answer:

The equation is $(x-5)^{2}+(y-20)^{2}=9$

Step-by-step explanation:

Given the following equation in the plane :

$(x-a)^{2}+(y-b)^{2}=R^{2}$

The set of points $(x,y)$ that satisfy the equation graph a circle in the plane.The circle is centered at $(a,b)$ and the radius of the circle is R.

For example, the set of points that satisfy : $(x-2)^{2}+(y-3)^{2}=9$

graph a circle of radius $\sqrt{9}=3$ centered at the point $(2,3)$

The first step to solve this exercise is to find the center of the circle.

The rectangle has vertices $(0,0),(0,40),(10,40)$ and $(10,0)$ so they will form a rectangle with width 10 units and height 40 units.

Given this situation, the center of the rectangle is at $(5,20)$ (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

$(x-5)^{2}+(y-20)^{2}=R^{2}$

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

$10units=200feet$ ⇒ $60feet=\frac{(60).(10)}{200}units=3units$

The radius of 60 feet is represented with 3 units in the blueprint.

Now we replace $R=3$ in the equation :

$(x-5)^{2}+(y-20)^{2}=3^{2}$ ⇒

$(x-5)^{2}+(y-20)^{2}=9$

And that is the equation of the circle.