The point \( (3, b) \) lies on the circle with radius 6 and center \( (-1,-1) \). What are the possible values of \( b \) ? \( b=\square \) Type an integer or decimal rounded to one decimal place as needed. Use a comma to separate answers as needed.)

To find the possible values of \( b \), we can use the equation of the circle. The general equation of a circle with center \( (h, k) \) and radius \( r \) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \] For this problem, the center is \( (-1, -1) \) and the radius is \( 6 \). Plugging in the center and radius, we have: \[ (x + 1)^2 + (y + 1)^2 = 6^2 \] This simplifies to: \[ (x + 1)^2 + (y + 1)^2 = 36 \] Now, substituting the point \( (3, b) \): \[ (3 + 1)^2 + (b + 1)^2 = 36 \] Calculating \( (3 + 1)^2 \): \[ 4^2 + (b + 1)^2 = 36 \] This leads to: \[ 16 + (b + 1)^2 = 36 \] Subtracting 16 from both sides gives: \[ (b + 1)^2 = 20 \] Now, solving for \( b + 1 \), we take the square root: \[ b + 1 = \sqrt{20} \quad \text{or} \quad b + 1 = -\sqrt{20} \] Calculating \( \sqrt{20} \): \[ b + 1 = 2\sqrt{5} \quad \text{or} \quad b + 1 = -2\sqrt{5} \] Subtracting 1 from both cases: \[ b = 2\sqrt{5} - 1 \quad \text{or} \quad b = -2\sqrt{5} - 1 \] Calculating the numerical values: 1. \( 2\sqrt{5} \approx 4.472 \), so \( b \approx 4.472 - 1 = 3.472 \) 2. \( -2\sqrt{5} \approx -4.472 \), so \( b \approx -4.472 - 1 = -5.472 \) Thus, rounded to one decimal place, the possible values of \( b \) are: \[ b \approx 3.5, -5.5 \] Final answer: \( b = 3.5, -5.5 \)