Exercise 43 Page 635

The standard form for the equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^{2} + (y - k)^{2} = r^{2} .$

We are already given the radius $r$ of each gear. The diagrams given will allow us to identify the centers of each circle, $(h, k) .$

Let's start with the largest gear with radius

6

inches. The center of the gear is at the origin

(0, 0) .

Let's substitute these values into the equation of a circle and simplify.

(x - 0)^{2} + (y - 0)^{2} = 6^{2} ⇕ x^{2} + y^{2} = 36

The equation for the gear with radius

6

is

x^{2} + y^{2} = 36 .

Let's find the equation for the gear with radius

4

inches. The center of the gear is at

(8, 6) .

Let's substitute these values into the equation of a circle and simplify.

(x - 8)^{2} + (y - 6)^{2} = 4^{2} ⇕ (x - 8)^{2} + (y - 6)^{2} = 16

The equation for the gear with radius

4

is

(x - 8)^{2} + (y - 6)^{2} = 16 .

Finally, let's find the equation for the gear with a radius of

2

inches. The center of the gear is at

(8, 0) .

Let's substitute these values into the equation of a circle and simplify.

(x - 8)^{2} + (y - 0)^{2} = 2^{2} ⇕ (x - 8)^{2} + y^{2} = 4