Question

in order to find $\frac{dy}{dt}$, differentiate both sides of the equation $y = \tan(35^{circ})x$. substitute $\frac{dx}{dt}=14$ km/hr into $\frac{dy}{dt}=\tan(35^{circ})\frac{dx}{dt}$ and solve for $\frac{dy}{dt}$. $\frac{dy}{dt}=\tan(35^{circ})\frac{dx}{dt}=\tan(35^{circ})(14$ km/hr)$approxsquare$ km/hr (round to two decimal places as needed.)

Explanation:

Step1: Identify given values

We are given $\frac{dx}{dt}=14$ km/hr and $y = \tan(35^{\circ})x$. Differentiating both sides with respect to $t$ using the chain - rule gives $\frac{dy}{dt}=\tan(35^{\circ})\frac{dx}{dt}$.

Step2: Substitute the value of $\frac{dx}{dt}$

Substitute $\frac{dx}{dt} = 14$ km/hr into the equation $\frac{dy}{dt}=\tan(35^{\circ})\frac{dx}{dt}$. We know that $\tan(35^{\circ})\approx0.7002$. Then $\frac{dy}{dt}=0.7002\times14$ km/hr.

Step3: Calculate the result

$\frac{dy}{dt}=0.7002\times14 = 9.8028\approx9.80$ km/hr.

Answer:

$9.80$ km/hr