To determine the equation of the line $l$ that passes through the point $P(4,7)$ and has an inclination angle $v = 30^\circ$, we can proceed as follows:

Step-by-Step Solution

1. Find the Slope (m): The inclination angle $v = 30^\circ$ gives us the slope $m$ of the line using: $m = \tan(v)$ Since $v = 30^\circ$, $m = \tan(30^\circ) = \frac{1}{\sqrt{3}}$

2. Use the Point-Slope Form of the Line: 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 $P(4, 7)$ and $m = \frac{1}{\sqrt{3}}$, we get: $y - 7 = \frac{1}{\sqrt{3}} (x - 4)$

3. Rewrite in Slope-Intercept Form (if needed): Expanding and rearranging terms to get the equation in $y = mx + b$ form: $y = \frac{1}{\sqrt{3}} x - \frac{4}{\sqrt{3}} + 7$ Simplifying further, we can leave it in this form or approximate values if desired.

Thus, the equation of the line $l$ is: $y = \frac{1}{\sqrt{3}} x + \left(7 - \frac{4}{\sqrt{3}}\right)$

Would you like further details or have any questions on this solution?

Additional Related Questions

1. What would the equation be if the line had passed through a different point, say $P(2,5)$, with the same angle?
2. How would the equation change if the angle was increased to $45^\circ$?
3. Can we rewrite the equation in standard form, $Ax + By = C$?
4. How would we find the x-intercept and y-intercept of this line?
5. What would the equation of a perpendicular line passing through $P(4,7)$ look like?

Tip

When working with inclination angles, remember that small angles (like 30°) produce slopes less than 1, while larger angles (approaching 90°) result in very steep slopes.