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)

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