1. Problem statement: Given a smoke stack centered at the origin with four radial labels C-4 (top-left), C-3 (top-right), C-2 (bottom-right) and C-1 (bottom-left) and radial lines at $\pm 45^\circ$, compute the planar $(x,y)$ coordinates for the listed repair points using the provided radial distances where available. 2. Coordinate mapping: By the diagram hint, use the mappings C-1 = bottom-left at angle $225^\circ$ and C-2 = bottom-right at angle $315^\circ$ measured from the positive $x$-axis. 3. Trigonometric formula: For a point at radius $r$ and angle $\theta$ the planar coordinates are given by the display formula below. $$x = r\cos\theta,\quad y = r\sin\theta$$ 4. Useful exact cosine and sine values at $45^\circ$ multiples: $ \cos 225^\circ = -\tfrac{\sqrt{2}}{2}$ \cos 315^\circ = \tfrac{\sqrt{2}}{2}$ \sin 225^\circ = -\tfrac{\sqrt{2}}{2}$ \sin 315^\circ = -\tfrac{\sqrt{2}}{2}$ 5. Compute Corner 1 - A with $r=3.7$ at $\theta=225^\circ$. Substitute into the formula to get the exact and numeric coordinates. $x = 3.7\cos 225^\circ = 3.7\left(-\tfrac{\sqrt{2}}{2}\right) = -\tfrac{3.7\sqrt{2}}{2}$ $y = 3.7\sin 225^\circ = 3.7\left(-\tfrac{\sqrt{2}}{2}\right) = -\tfrac{3.7\sqrt{2}}{2}$ Numeric approximation rounded to four decimals gives the final coordinate for Corner 1 - A as $(-2.6163,\ -2.6163)$. 6. Compute Corner 2 - B with $r=3.7$ at $\theta=315^\circ$. $x = 3.7\cos 315^\circ = 3.7\left(\tfrac{\sqrt{2}}{2}\right) = \tfrac{3.7\sqrt{2}}{2}$ $y = 3.7\sin 315^\circ = 3.7\left(-\tfrac{\sqrt{2}}{2}\right) = -\tfrac{3.7\sqrt{2}}{2}$ Numeric approximation rounded to four decimals gives the final coordinate for Corner 2 - B as $(2.6163,\ -2.6163)$. 7. Compute Corner 1 - C with $r=3.5$ at $\theta=225^\circ$. $x = 3.5\cos 225^\circ = 3.5\left(-\tfrac{\sqrt{2}}{2}\right) = -\tfrac{3.5\sqrt{2}}{2}$ $y = 3.5\sin 225^\circ = 3.5\left(-\tfrac{\sqrt{2}}{2}\right) = -\tfrac{3.5\sqrt{2}}{2}$ Numeric approximation rounded to four decimals gives the final coordinate for Corner 1 - C as $(-2.4749,\ -2.4749)$. 8. Compute Corner 2 - C with $r=3.5$ at $\theta=315^\circ$. $x = 3.5\cos 315^\circ = 3.5\left(\tfrac{\sqrt{2}}{2}\right) = \tfrac{3.5\sqrt{2}}{2}$ $y = 3.5\sin 315^\circ = 3.5\left(-\tfrac{\sqrt{2}}{2}\right) = -\tfrac{3.5\sqrt{2}}{2}$ Numeric approximation rounded to four decimals gives the final coordinate for Corner 2 - C as $(2.4749,\ -2.4749)$. 9. Corner 1 - D (labelled "3rd ftr") has no explicit radial distance provided in the notes, so the planar coordinates cannot be numerically computed without the radius value. Provide the formula for use if a radius $r$ is given: $x = r\cos 225^\circ = -\tfrac{r\sqrt{2}}{2}$ and $y = r\sin 225^\circ = -\tfrac{r\sqrt{2}}{2}$. 10. Corner 2 - D (labelled "3rd ftr") likewise lacks an explicit radial distance, so numerical coordinates cannot be computed without the radius value. Provide the formula for use if a radius $r$ is given: $x = r\cos 315^\circ = \tfrac{r\sqrt{2}}{2}$ and $y = r\sin 315^\circ = -\tfrac{r\sqrt{2}}{2}$. 11. Final summary of computed numerical coordinates for the items with provided radii. Corner 1 - A (3.7) : $(-2.6163,\ -2.6163)$ Corner 2 - B (3.7) : $(2.6163,\ -2.6163)$ Corner 1 - C (3.5) : $(-2.4749,\ -2.4749)$ Corner 2 - C (3.5) : $(2.4749,\ -2.4749)$ Corner 1 - D (3rd ftr) : radius not provided, coordinate formulas given above for substitution. Corner 2 - D (3rd ftr) : radius not provided, coordinate formulas given above for substitution.