Question

a circular walkway is to be built around a monument, with the monument as the center. the distance from the monument to any point on the inner boundary of the walkway is 30 feet. concentric circles a. what is the equation of the inner boundary of the walkway? use a coordinate system with the monument at (0,0). b. if the walkway is 7 feet wide, what is the equation of the outer boundary of the walkway? question help: ebook submit question

Explanation:

Step1: Recall the standard - form of a circle equation

The standard form of the equation of a circle with center \((h,k)\) and radius \(r\) is \((x - h)^2+(y - k)^2=r^2\). Here, the center of the circle (the location of the monument) is \((h,k)=(0,0)\), and for the inner - boundary, the radius \(r = 30\).

Step2: Substitute the values into the equation

Substituting \(h = 0\), \(k = 0\), and \(r = 30\) into the standard - form equation, we get \((x-0)^2+(y - 0)^2=30^2\), which simplifies to \(x^{2}+y^{2}=900\).

Step3: Find the radius of the outer - boundary

The width of the walkway is 7 feet. So the radius of the outer - boundary \(R=r + 7=30+7 = 37\) feet.

Step4: Write the equation of the outer - boundary

Using the standard - form of the circle equation \((x - h)^2+(y - k)^2=R^2\) with \(h = 0\), \(k = 0\), and \(R = 37\), we get \((x-0)^2+(y - 0)^2=37^2\), which simplifies to \(x^{2}+y^{2}=1369\).

Answer:

a. \(x^{2}+y^{2}=900\)
b. \(x^{2}+y^{2}=1369\)