The three sides and diagonal of the rectange above are strictly increasing with time. At the instant when $x=4$ and $y=3$, $\dfrac{dx}{dt}=\dfrac{dz}{dt}$ and $\dfrac{dy}{dt}=k\dfrac{dz}{dt}$. What is the value of $k$ at that instant?

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The relationship between the three variables can be described using the pythagorean theorem:

$$x^2+y^2=z^2$$

If we take the derivative of both sides with respect to time, we get:

$$2x\frac{dx}{dt}+2y\frac{dy}{dz} = 2z\frac{dz}{dt}$$

Simplify by dividing each term by 2.

$$x\frac{dx}{dt}+y\frac{dy}{dz} = z\frac{dz}{dt}$$

We can substitute values for $\dfrac{dx}{dt}$ and $\dfrac{dy}{dt}$ so that we have an equation with only $\dfrac{dz}{dt}$.

$$x\left(\frac{dz}{dt}\right)+y\left(k\frac{dz}{dt}\right)=z\frac{dz}{dt}$$

Simplify by dividing each term by $\dfrac{dz}{dt}$.

$$x+ky=z$$

At the instant $x=4$ and $y=3$, the hypotenuse is $z=5$. (3, 4, 5 is a common pythagorean triplet)

$$4+k(3)=5$$ $$3k = 1$$ $$k = \boxed{\frac{1}{3}}$$