To determine the equation of the line that passes through the point $(0, 4)$ and makes an angle of $30^\circ$ with the positive direction of the x-axis, we can use the following steps:

Step-by-Step Solution

1. Find the slope of the line:

   The slope $m$ of a line that makes an angle $\theta$ with the positive x-axis is given by:

   $m = \tan(\theta)$

   Given that $\theta = 30^\circ$:

   $m = \tan(30^\circ) = \frac{1}{\sqrt{3}}$

2. Use the point-slope form of the line equation:

   The point-slope form of a line passing through a point $(x_1, y_1)$ with slope $m$ is:

   $y - y_1 = m(x - x_1)$

   Substituting the point $(0, 4)$ and the slope $m = \frac{1}{\sqrt{3}}$:

   $y - 4 = \frac{1}{\sqrt{3}}(x - 0)$

   Simplifying:

   $y = \frac{1}{\sqrt{3}}x + 4$

3. Convert to standard form (if needed):

   The equation of the line in slope-intercept form is:

   $y = \frac{1}{\sqrt{3}}x + 4$

   This is the equation of the line passing through the point $(0, 4)$ and making an angle of $30^\circ$ with the positive x-axis.

Final Answer

The equation of the line is:

$y = \frac{1}{\sqrt{3}}x + 4$

Would you like more details or have any other questions?

Related Questions:

1. What is the general form of the equation for this line?
2. How would the equation change if the line passed through a different point, say $(1, 3)$?
3. What is the y-intercept and x-intercept of this line?
4. How do you find the equation of a line if it makes a different angle, like $45^\circ$ with the x-axis?
5. Can you determine the distance between the origin and the line?

Tip:

When given an angle with the x-axis, always use the tangent of that angle to determine the slope of the line.