Eric and Harrison are standing in a field, and Eric is 400 feet directly East of Harrison. Eric starts to walk North at a rate of 4 feet per second, while Harrsion starts to walk South at the same time at a rate of 6 feet per second. After 30 seconds, at what rate is the distance between Eric and Harrison changing?
We can model the rate of a change as a right triangle with base \(x = 400\) feet and height y increasing at a rate of 10 feet per second.
After 30 seconds, \(y=10\cdot30=300\)
If the distance between Eric and Harrison is z, the Pythagorean Theorem gives us: \(z^2 = x^2 + y^2 \).
The distance between Eric and Harrison is: \(x^2=300^2+400^2\Rightarrow x=500\)
Differentiating both sides also yields \(2z · z' = 2x · x' + 2y · y'\). Plugging in our values, we get \(2 · 500z' = 2 · 400 · 0 + 2 · 300 · 10\), which gives us \( z' = 6\) feet per second.
I hope this helped,