The pictures above describes the question. We have to find the rate of change in x-axis direction.
The answer is derived from implicit differentiation and is $4/3$. The process is: [y(t) gives y-axis length (red line) and x(t) gives x-axis length (green line) where t is seconds after initial picture]
$$ \begin{align} &(y(t))^2 + (x(t))^2 &= 5^2 \\ &\implies \frac {d[(y(t))^2 + (x(t))^2]} {dt} &= \frac{d25} {dt} \\ &\implies \frac {2y\frac{dy(t)}{dt} + 2x\frac{dx(t)}{dt}} {dt} &= 0 \\ &\implies {2y\frac{dy(t)}{dt} + 2x\frac{dx(t)}{dt}} &= 0 \\ \end{align} $$
At $t=0$,
$$ \begin{align} &{2(4)(-1) + 2(3)\frac{dx(t)}{dt}} &= 0 \\ &\implies -8+6\frac{dx(t)}{dt} &= 0 \\ &\implies \frac{dx(t)}{dt} &= \frac 86 \\ &\implies \frac{dx(t)}{dt} &= \frac 43 \\ \end{align} $$
But by solving intuitionally: $y(t)$ becomes 0 after $4/1 = 4$ seconds so the $x(t) = 5$ when $t = 4$. Since the rate of change is constant then we can calculate rate of change as change in $x(t)$ from $t=0$ to $t=4$ divided by $4-0=4$
$$ \text{Rate of change = } = \frac {5-3} {4} = \frac 2 4 = \frac 12 $$
This contradicts above result. So which one is correct and why?
One possible answer I thought could be that the rate of change of $x(t)$ is not constant as $y(t)$ is. So is that it?