If a right triangle with hypotenuse of length $26$ has a leg of length $10$; and this leg (of length 10) is constantly incremented at a speed of $4$ units per second (let "seconds" be considered as another variable t) such that the hypotenuse and right angle are kept constant, at what speed will the remaining leg decrease?
First of all, the remaining leg would, by the Pythagorean Theorem, be of length $24$.
If I solve this as a related rates problem, I would get that, since $x^2 + y^2 = z^2$ (let $x = 10$, $y = 24$, $z = 26$), taking the derivative of both sides with respect to $t$ we have: $2x$$dx\over dt$$ + 2y$$dy\over dt$$ = $$2z$$dz\over dt$. Since $z$ is to be kept constant, we have, algebraically: $dy\over dt$$ = –$$x\over y$$dx\over dt$. Substituting $dx\over dt$$ = 4$ and the given intial $x = 10$, $y = 24$ we get $dy\over dt$$ ≈ –1.667$.
Now, to put this to the test, I decided to see if this speed stays constant. Let one unit of $t$ be incremented. Then, $x$ shall be incremented by $4$, and $y$ by approximately $–1.667$. So $x$ shall become $14$, and $y$ approximately shall be $22.334$. However, when I substitute the $x$ and $y$ to our previous equation $dy\over dt$$ = –$$x\over y$$dx\over dt$. I get, approximately, $dy\over dt$$ = 2.51$. What is going on here? What flaw am I permitting? I believe the speed in which $y$ decreases is supposed to stay constant, but this is not at all what I am observing. Could anyone please point out where I'm making a fallacy? Thank you im advance.
Your intuition that the rate of change of $y$ should be constant when $x$ changes at a constant rate is wrong.
To see why, imagine the extreme cases. When $x$ is very small, $y$ is just a little less than the hypotenuse. The leg of length $x$ is nearly perpendicular to hypotenuse, so when it moves it hardly drags the far vertex towards it at all, and the other leg $y$ decreases slowly.
When $x$ is nearly as long as the hypotenuse that leg and the hypotenuse are nearly parallel. Changing $x$ at a constant rate shortens the other leg rapidly. In fact that rate of decrease increases without bound.