Assume a quarter, diameter 0.96 inches, is centered at the point left parenthesis negative 2 comma 5 right parenthesis where each unit represents one inch. Write the equation of a circle that would precisely enclose the coin.

Question

(1 point)
Responses

left parenthesis x plus 2 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 0.2304
Image with alt text: left parenthesis x plus 2 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 0.2304

left parenthesis x minus 2 right parenthesis squared plus left parenthesis y plus 5 right parenthesis squared equals 0.2304
Image with alt text: left parenthesis x minus 2 right parenthesis squared plus left parenthesis y plus 5 right parenthesis squared equals 0.2304

left parenthesis x plus 2 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 0.9216
Image with alt text: left parenthesis x plus 2 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 0.9216

left parenthesis x plus 2 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 0.96

Answer 1

To write the equation of a circle that encloses a quarter with a diameter of 0.96 inches, we first need to find the radius of the circle. The radius \( r \) is half of the diameter, so:

\[ r = \frac{0.96}{2} = 0.48 \text{ inches} \]

The standard equation of a circle centered at the point \((h, k)\) is:

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

In this case, the center of the circle is given as \((-2, 5)\). Therefore, \( h = -2 \) and \( k = 5\).

Now, we can substitute the center and the radius into the equation:

1. First, we need to calculate \( r^2 \): \[ r^2 = (0.48)^2 = 0.2304 \]

2. Now we can substitute \( h \), \( k \), and \( r^2 \) into the equation: \[ (x + 2)^2 + (y - 5)^2 = 0.2304 \]

Thus, the equation of the circle that precisely encloses the coin is:

\[ \boxed{(x + 2)^2 + (y - 5)^2 = 0.2304} \]