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

Answer:

C. 3

Explanation:

Step1: Apply Pythagorean theorem

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

Step2: Differentiate with respect to time $t$

Using the chain - rule, we have $2z\frac{dz}{dt}=2x\frac{dx}{dt}+2y\frac{dy}{dt}$.

Step3: Find the value of $z$

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

Step4: Substitute given values

Since $\frac{dx}{dt}=\frac{dz}{dt}$, the equation $z\frac{dz}{dt}=x\frac{dx}{dt}+y\frac{dy}{dt}$ becomes $5\frac{dz}{dt}=4\frac{dz}{dt}+3k\frac{dz}{dt}$.

Step5: Solve for $k$

Cancel out $\frac{dz}{dt}$ (since $\frac{dz}{dt}\neq0$ as the sides and diagonal are increasing) and get $5 = 4+3k$. Then $3k=1$, so $k = 3$.