Find the miniumum distance between A and B?

Question

Two particles, A and B, are in motion in the xy-plane. Their coordinates at each instant of time (t >= 0) are given by x_A = t, y_A = t^2, x_B = 2*t, and y_B = 2. Find the minimum distance between A and B. Please round your answers to two decimal places
(Studying for my Calculus midterm! This is a problem from the book, can't figure it out. Thank you!)

Answer 1

A = (t, t^2), B = (2t, 2)

Distance between them is D(t) = sqrt[t^2 + (2 - t^2)^2 ]

From this point, you would find the first derivative using chain rule and power rule and find all the values of t for which D ' (t) = 0.

After that, to find the critical points that are minima, you would find the second derivative and determine the values of t for which D " (t) > 0.

Answer 2

By Pythagorean theorem, the distance AB between the two points at time t should be
sqrt [(t - 2*t)^2 + (t^2 - 2)^2]
So then differentiate this expression and set equal to zero and solve for t, and then substitute that value of t back into the distance expression to get AB