Determine the center and radius of the following circle equation:
x^2 + y^2 + 12x + 16y + 19 = 0

radius: 9

center: (-6, -8)

Step-by-step explanation:

First, you may wan to rearrange the equation so it is easier to sole, like:

x^2 + 12x + y^2 + 16y + 19 =0

Next, we need to complete the square. x^2 + 12x is the start of the square of x + 6, which ends in 36, and y^2 + 16y is the start of the square of x + 8, which ends in 64. We can start to write in the squares like:

x^2 + 12x + 36 - 36 + y^2 + 16y + 64 - 64 + 19 = 0

(36 & 64 are subtracted so the equation stays the same)

You can factor getting:

(x + 6)^2 - 36 + (y + 8)^2 - 64 + 19 = 0

You can combine the constants to get -36 - 64 + 19 = -81 and we can add 81 to both sides to get:

(x + 6)^2 + (y + 8)^2 = 81

The standard form of a circle equation is:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center and r is the radius.

In this case, by substituting, we get the center is (-6, -8) and the radius is 9.

answer: the price of one pound cabbage = $ 0.8

the point (10,8) represents that the cost of 10 pounds cabbage is $ 8

step-by-step explanation:

since, here is a proportional relation between the number of pounds of cabbage (x) and the price in dollars (y).

therefore, we can write,

⇒ y = k x

where k is the constant of proportionality.

now, point (5,4) is in the line.

thus, it will satisfies the above equation of line.

⇒ 4 = 5 k

⇒ k = 0.8

thus, the equation of the line is,

y = 0.8 x

a) for x = 1,

y = 0.8

therefore, the cost of one pound of cabbage is 0.8 dollars.

b) in the below graph (10,8) represents that the cost of 10 pounds of cabbage is 8 dollars.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4