We are tasked with finding the point in the first quadrant where the line $y = 0.5x + 3$ intersects the circle with radius 3 and center at $(0, 3)$. The equation of the circle is derived from the standard form of a circle, which is:

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

where $(h, k)$ is the center, and $r$ is the radius. For our circle, the center is $(0, 3)$ and the radius is 3, so the equation of the circle becomes:

$x^2 + (y - 3)^2 = 9$

Step 1: Substitute the equation of the line into the equation of the circle

We substitute $y = 0.5x + 3$ into the equation of the circle:

$x^2 + ((0.5x + 3) - 3)^2 = 9$

Simplifying the equation:

$x^2 + (0.5x)^2 = 9$

$x^2 + 0.25x^2 = 9$

$1.25x^2 = 9$

Step 2: Solve for $x$

$x^2 = \frac{9}{1.25} = 7.2$

$x = \sqrt{7.2} \approx 2.683$

Step 3: Find the corresponding $y$-value

Now, substitute $x \approx 2.683$ into the equation of the line $y = 0.5x + 3$:

$y = 0.5(2.683) + 3 \approx 1.341 + 3 = 4.341$

Conclusion:

The point of intersection in the first quadrant is approximately $(2.683, 4.341)$.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. What is the general formula for the intersection of a line and a circle?
2. How can we determine if a line intersects a circle at more than one point?
3. What is the geometric significance of a circle's center lying on a line?
4. How would the solution change if the radius of the circle increased?
5. How would the intersection points change if the slope of the line were steeper?

Tip: When solving for intersections of curves, always verify if multiple solutions exist by considering the discriminant of the quadratic equation formed.