Question 6. The arch of a bridge over a river is modelled by the formula \( h=-\frac{1}{40}(x-25)^{2}+15.625 \), where \( h \) is the height of the span above the river, and \( d \) is the distance from the left bank of the river. A horizontal beam runs from one side of the arch to the other at a height of 10 m above the river. The length of this beam is closest to

To find the length of the horizontal beam that runs at a height of 10 m above the river, we first set the equation of the arch equal to 10. Thus, we solve \[ -\frac{1}{40}(x-25)^{2}+15.625 = 10. \] Rearranging gives us \( -\frac{1}{40}(x-25)^{2} = -5.625 \). Multiplying by -40 leads to \[ (x-25)^{2} = 225. \] Taking the square root yields \( x - 25 = 15 \) or \( x - 25 = -15 \), leading to the solutions \( x = 40 \) and \( x = 10 \). Therefore, the distance along the beam is \[ 40 - 10 = 30. \] The length of the beam is closest to 30 m. --- The formula of the bridge arch is a classic example of a quadratic function, which often models real-world phenomena like bridges and fountains. Quadratics are characterized by their parabolic shape, making them ideal for structures needing wind resistance and stability. The vertex form of the equation reveals the highest point, or vertex, of the arch, essential for understanding the design's structure and strength. When working with heights in real-world situations, always check if the prescribed height (like the 10 m for the beam) can be reached by substituting back into the function. It's a common mistake to neglect ensuring the height falls within the defined bounds of the function, which can lead to inaccurate conclusions about feasibility in engineering problems.