### Home > APCALC > Chapter 10 > Lesson 10.1.4 > Problem10-44

10-44.

In a baseball game, as Barry hits a home run, Sammy leaves 1st base for 2nd base, running at $12$ ft/sec. At the same time, Mark leaves 2nd base for 3rd base, running at $16$ ft/sec. If the bases are $90$ feet apart, at what amount of time after Barry hit the ball is the distance between Sammy and Mark at a minimum? What is the distance between them at that time? .

Sammy's distance is the horizontal leg of a right triangle. It is decreasing with time.
Write an expression for Sammy's distance (from second base).

Mark's distance is the vertical leg of a right triangle. It is increasing with time.
Write an expression for Mark's distance (from third base).

Use the Pythagorean Theorem to write an equation for the distance between Sammy and Mark.

Differentiate your equation and set it equal to 0 to solve for the time when the distance between Sammy and Mark is a minimum.