Ailani draws a map of her local town. She places the town hall at the origin of a coordinate plane and represents a lake
with a circle drawn on the map. The center of the lake is 19 miles east and 3 miles south of the town hall, and the
radius of the lake is 0.5 miles.
If the positive x-axis represents east and the positive y-axis represents north, which equation represents the lake?
O (x + 19)2 + (y - 3)2 = 0.5
O (x – 19)2 + (y + 3)2 = 0.5
O (x + 19)2 + y - 3)2 = 0.25
O (x - 19)2 + ( + 3)2 = 0.25
T

The equation of the lake with a center 19 miles east and 3 miles south with radius of 0.5 miles is (x + 19)² + (y + 3)² = 0.25

An equation is an expression that shows the relationship between two or more variables and numbers.

The equation for a circle is given as:

(x - h)² + (y - k)² = r²

Where (h, k) is the center of the circle and r is the radius.

The center of the lake is 19 miles east and 3 miles south, i.e. (-19, -3) and the radius is 0.5 miles.

The equation of the lake is:

(x - (-19))² + (y - (-3))² = 0.5²

(x + 19)² + (y + 3)² = 0.25

The equation of the lake with a center 19 miles east and 3 miles south with radius of 0.5 miles is (x + 19)² + (y + 3)² = 0.25

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Answer: D.  [tex](x-19)^2+(y+3)^2=0.25[/tex]

Step-by-step explanation:

Let's review what we know:

A circle can be depicted on the coordinate plane with the equation [tex](x-h)^2+(y-k)=r^2[/tex], where:

19 miles east of the origin would be 19 units to the right of the origin on the coordinate plane, as east is the rightwards direction in real life. In short, h = 19.

Let's plug in that value: [tex](x-(19))^2+(y-k)^2=r^2[/tex].

So we know that the x-coordinate defining part of the equation would have to be [tex](x-19)^2[/tex].

Meanwhile, 3 miles south of the origin would be modeled as three units underneath it, because south would be considered "downwards". The y-value (k) would be negative 3.

Putting what we have so far together, the center of the circle would be (19, -3).

Therefore, the y-coordinate defining part of the equation would be [tex](y+3)^2[/tex].

Here's what the equation looks like with the work we've completed so far: [tex](x-19)^2+(y+3)^2=r^2[/tex]

Finally, let's plug in the radius value 0.5.

0.5 * 0.5 = 0.25

Ailani would use the equation [tex](x-19)^2+(y+3)^2=0.25[/tex] to model the lake in relation to the town hall. I hope that makes sense!

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