Tunnel Bus Height B37Ec6

1. **State the problem:** We have a tunnel with a semicircular cross-section of diameter $10$ m. A bus is $2$ m from one side of the tunnel, and its roof just touches the tunnel's roof. We need to find the height of the bus. 2. **Identify the radius and coordinate setup:** The radius of the semicircle is half the diameter: $$r = \frac{10}{2} = 5 \text{ m}$$ 3. **Equation of the semicircle:** Place the semicircle centered at the origin $(0,0)$ with radius $5$. The semicircle is above the $x$-axis, so its equation is: $$y = \sqrt{r^2 - x^2} = \sqrt{25 - x^2}$$ 4. **Position of the bus:** The bus is $2$ m from one side. Since the tunnel extends from $-5$ to $5$ on the $x$-axis, the bus is at: $$x = -5 + 2 = -3$$ 5. **Calculate the height of the tunnel at $x = -3$:** $$y = \sqrt{25 - (-3)^2} = \sqrt{25 - 9} = \sqrt{16} = 4$$ 6. **Interpretation:** The height of the tunnel at the bus position is $4$ m, so the bus height is $4$ m. **Final answer:** $$\boxed{4 \text{ meters}}$$