Question

question: beginning 156 miles directly east of the city of uniontown, a truck travels due south. if the truck is travelling at a speed of 31 miles per hour, determine the rate of change of the distance between uniontown and the truck when the truck has been travelling for 81 miles. (do not include units in your answer, and round to the nearest tenth.) provide your answer below:

Explanation:

Step1: Define variables

Let $x = 156$ (initial east - west distance), $y$ be the south - north distance, $z$ be the distance between the truck and Uniontown. By Pythagorean theorem $z=\sqrt{x^{2}+y^{2}}$.

Step2: Differentiate with respect to time $t$

$2z\frac{dz}{dt}=2y\frac{dy}{dt}$. Given $\frac{dy}{dt}=31$ mph and when $y = 81$, $z=\sqrt{156^{2}+81^{2}}=\sqrt{24336 + 6561}=\sqrt{30897}\approx175.8$.

Step3: Solve for $\frac{dz}{dt}$

$\frac{dz}{dt}=\frac{y}{z}\cdot\frac{dy}{dt}=\frac{81}{175.8}\times31\approx14.4$.

Answer:

$14.4$