- Examine the table of given points \((x, f(x))\) representing the roof slope.
- Compare the vertical height values \(f(x)\) to find the minimum value.
- Identify the horizontal distance \(x\) corresponding to this minimum vertical height.
- Conclude that the drain is located at \(\boxed{80}\) feet from the wall.
Explanation
- Problem Analysis
The architectural drawing provides a relationship between the horizontal distance from a wall (the y-axis) and the vertical height of a roof. The given data points are:
\((0, 8), (20, 6), (40, 4), (80, 0), \text{ and } (160, 8)\).
The problem states that the roof slopes to a drain located at the minimum point of the function \(f(x)\). To find the location of this drain, we must identify the minimum value of \(f(x)\) within the provided data set.
- Identifying the Minimum Point
We observe the values of \(f(x)\) provided in the table:
- At \(x = 0\), \(f(x) = 8\)
- At \(x = 20\), \(f(x) = 6\)
- At \(x = 40\), \(f(x) = 4\)
- At \(x = 80\), \(f(x) = 0\)
- At \(x = 160\), \(f(x) = 8\)
Comparing these values, the minimum value is \(0\), which occurs when the horizontal distance \(x\) is \(80\) feet. Since the drain is placed at this minimum height, the drain is located exactly \(80\) feet from the wall.
- Final Answer
Based on the data, the drain is located at the point where \(f(x)\) reaches its minimum value of \(0\). The corresponding horizontal distance \(x\) is \(80\) feet. Therefore, the distance from the wall (the y-axis) is \(\boxed{80}\) feet.
Examples
In architectural and structural engineering, determining the precise location of a drain on a sloped roof is critical for effective water management. By mapping the elevation (height) of the roof surface as a function of the distance from the supporting walls, engineers can use data points to ensure that water flows toward the lowest point. This simple analysis of finding a minimum value is fundamental in designing roofs that prevent water pooling and protect building integrity.