Question

A water reserve follows the path given by y = 2x + 20, where each unit represents 1 kilometer. A town is located at (50, 5). The town’s laws require that the water reserve be the shortest possible distance from the town. How close does the water reserve come to the town?

Answer 1

To find the shortest distance from the town to the water reserve, we calculate the perpendicular distance from the point (50,5) to the line y = 2x + 20, which involves finding the intersection of the water reserve's path and a perpendicular line passing through the town.

Step-by-step explanation:

The question involves finding the shortest distance from a point to a line in a coordinate plane, which is a common problem in geometry and algebra. To solve this, we need to determine the perpendicular distance from the given point (50, 5) to the water reserve's path given by the equation y = 2x + 20. To do this, we first need to find the slope of the line perpendicular to the water reserve's path, which will be the negative reciprocal of the slope of the given line. Since the slope of the water reserve's path is 2, the slope of the perpendicular line is -1/2. We then write the equation of the line passing through the town and having this slope. Once we have the equation of the perpendicular line, we can find the point of intersection between this line and the water reserve's path. The distance between the town and this point of intersection will give us the shortest distance between the town and the water reserve.