To find the equation of the straight line that makes an angle of 30° with the positive x-axis and cuts the y-axis at y = 5, we can follow these steps: ### Step 1: Understand the Slope of the Line The slope (m) of a line that makes an angle θ with the positive x-axis can be calculated using the tangent function: \[ m = \tan(\theta) \] For our case, θ = 30°: \[ m = \tan(30°) = \frac{1}{\sqrt{3}} \quad \text{(since } \tan(30°) = \frac{1}{\sqrt{3}} \text{)} \] ### Step 2: Use the Point-Slope Form of the Line The point-slope form of a line's equation is given by: \[ y - y_1 = m(x - x_1) \] where (x₁, y₁) is a point on the line. We know that the line cuts the y-axis at y = 5, which means the point (0, 5) is on the line. Thus, we can substitute (x₁, y₁) = (0, 5) into the equation: \[ y - 5 = \frac{1}{\sqrt{3}}(x - 0) \] This simplifies to: \[ y - 5 = \frac{1}{\sqrt{3}}x \] ### Step 3: Rearranging the Equation Now, we can rearrange this equation to express y in terms of x: \[ y = \frac{1}{\sqrt{3}}x + 5 \] ### Step 4: Clear the Denominator To eliminate the square root in the denominator, we can multiply the entire equation by √3: \[ \sqrt{3}y = x + 5\sqrt{3} \] ### Step 5: Final Equation This gives us the final equation of the line: \[ \sqrt{3}y = x + 5\sqrt{3} \] ### Conclusion Now, let's compare this with the options provided: - **Option A:** \(y = \frac{x}{10} + 5\) - This is incorrect because the slope does not match. - **Option B:** \(y = x + 5\) - This is incorrect because the slope is 1, not \(\frac{1}{\sqrt{3}}\). - **Option C:** \(\sqrt{3}y = -x + 5\sqrt{3}\) - This is incorrect because the x-term has a negative sign, which does not match our derived equation. - **Option D:** \(\sqrt{3}y = x + 5\sqrt{3}\) - This is correct as it matches our derived equation. ### Summary of Key Points - The slope of a line making a 30° angle with the x-axis is \(\frac{1}{\sqrt{3}}\). - The point-slope form of the line is used to derive the equation based on a known point. - The final equation can be simplified and rearranged to match standard forms. - Always check each option against the derived equation to confirm correctness. Thus, the correct answer is **D: \(\sqrt{3}y = x + 5\sqrt{3}\)**.