1. **State the problem:** We have a parabolic arc with a vertical axis, 10 m high and 5 m wide at the base. We want to find the width of the arc 2 m from the vertex. 2. **Set up the coordinate system and equation:** Place the vertex of the parabola at the origin $(0,0)$ with the parabola opening downward (since the arc is 10 m high). The height corresponds to $y=10$ at the base. 3. **General form of the parabola:** Since the axis is vertical, the parabola can be written as $$y = ax^2 + k$$ where $a$ is a constant and $k$ is the vertex's y-coordinate. Here, vertex at origin means $k=0$, so $$y = ax^2$$. 4. **Use given dimensions:** The arc is 10 m high, so the base is at $y=10$. The width at the base is 5 m, so the half-width is $2.5$ m. At $y=10$, $x=\pm 2.5$. 5. **Find $a$:** Substitute $x=2.5$, $y=10$ into $y = ax^2$: $$10 = a(2.5)^2 = a \times 6.25$$ $$a = \frac{10}{6.25} = 1.6$$ 6. **Equation of parabola:** $$y = 1.6x^2$$ 7. **Find width 2 m from vertex:** At $y=2$, solve for $x$: $$2 = 1.6x^2$$ $$x^2 = \frac{2}{1.6} = 1.25$$ $$x = \pm \sqrt{1.25} = \pm 1.118$$ 8. **Width at $y=2$:** The total width is twice the absolute value of $x$: $$\text{width} = 2 \times 1.118 = 2.236$$ meters. **Final answer:** The width of the arc 2 m from the vertex is approximately $2.24$ meters.