11 multiple choice 1 point the sides and diagonal of the rectangle above are strictly increasing with time…

Explanation:

Step1: Apply Pythagorean theorem

By the Pythagorean theorem, $x^{2}+y^{2}=z^{2}$.

Step2: Differentiate both sides with respect to $t$

Differentiating gives $2x\frac{dx}{dt}+2y\frac{dy}{dt}=2z\frac{dz}{dt}$. Divide through by 2: $x\frac{dx}{dt}+y\frac{dy}{dt}=z\frac{dz}{dt}$.

Step3: Find the value of $z$

When $x = 4$ and $y = 3$, by the Pythagorean theorem $z=\sqrt{4^{2}+3^{2}}=\sqrt{16 + 9}=5$.

Step4: Substitute given values

Since $\frac{dx}{dt}=\frac{dz}{dt}$ and $\frac{dy}{dt}=k\frac{dz}{dt}$, substitute into $x\frac{dx}{dt}+y\frac{dy}{dt}=z\frac{dz}{dt}$. We get $x\cdot1 + y\cdot k=z\cdot1$. Substitute $x = 4,y = 3,z = 5$: $4+3k = 5$.

Step5: Solve for $k$

Subtract 4 from both sides: $3k=5 - 4=1$. Then $k=\frac{1}{3}$.

Answer:

B. $\frac{1}{3}$