question a van is traveling due north at a speed of 80 km/h. if the van started off 6 km directly east of…

Explanation:

Step1: Define variables and relation

Let $y$ be the distance the van has traveled north and $x = 6$ km (constant east - west distance). We know that $\tan\theta=\frac{y}{x}$. Since $x = 6$, we have $\tan\theta=\frac{y}{6}$.

Step2: Differentiate both sides with respect to time $t$

Using the chain - rule, $\sec^{2}\theta\frac{d\theta}{dt}=\frac{1}{6}\frac{dy}{dt}$.

Step3: Find $\sec^{2}\theta$ when $y = 8$

First, when $y = 8$ and $x = 6$, by the Pythagorean theorem, the hypotenuse $r=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=10$. Then $\cos\theta=\frac{x}{r}=\frac{6}{10}=\frac{3}{5}$, and $\sec\theta=\frac{5}{3}$, so $\sec^{2}\theta=\frac{25}{9}$.

Step4: Substitute known values

We know that $\frac{dy}{dt}=80$ km/h. Substituting $\sec^{2}\theta=\frac{25}{9}$ and $\frac{dy}{dt}=80$ into $\sec^{2}\theta\frac{d\theta}{dt}=\frac{1}{6}\frac{dy}{dt}$, we get $\frac{25}{9}\frac{d\theta}{dt}=\frac{1}{6}\times80$.

Step5: Solve for $\frac{d\theta}{dt}$

Cross - multiply: $25\times6\times\frac{d\theta}{dt}=9\times80$. Then $150\frac{d\theta}{dt}=720$. So $\frac{d\theta}{dt}=\frac{720}{150}=\frac{24}{5}$ radians per hour.

Answer:

$\frac{24}{5}$