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). How close does the pipeline come to the town?

a) 70 kilometers
b) 45 kilometers
c) 5 kilometers
d) 25 kilometers

Answer 1

To calculate the closest distance between the town at (50, 5) and the water reserve path y = 2x + 20, the perpendicular distance formula is used, which results in approximately 51.44 kilometers, though this does not match the provided answer choices.

Step-by-step explanation:

To find the closest distance between the town at (50, 5) and the water reserve following the path y = 2x + 20, we need to calculate the perpendicular distance from the point to the line. This can be done using the point-to-line distance formula: d = |Ax + By + C| / √(A^2 + B^2), where A, B, and C are the coefficients from the line equation Ax + By + C = 0, adjusted to match our linear equation.

First, we rewrite the equation of the water reserve path in standard form: y = 2x + 20 to 2x - y + 20 = 0. Here, A = 2, B = -1, and C = 20. Substituting the town's coordinates (50, 5) and the coefficients into the distance formula, we get:

d = |2*50 - 1*5 + 20| / √(2^2 + (-1)^2) = |100 - 5 + 20| / √(4 + 1) = |115| / √5 = 115 / √5 ≈ 51.44

The closest distance is therefore approximately 51.44 kilometers. None of the given options matches this value exactly, so there may be an error in the provided options or a misunderstanding in the question. However, if we must choose from the given options, the closest value to the calculated distance is 45 kilometers (option b), but this is not the precise answer.