## Calculus (3rd Edition)

$75.89\,km/h$
Let the distance between the automobile and the farmhouse be $h$ and the distance traveled by the automobile past the intersection of the highway and the road be $x$. Then, using Pythagoras' theorem, we have $2^{2}+x^{2}= h^{2}$ When $x=6\,km$, $h=\sqrt {2^{2}+6^{2}}= 2\sqrt {10}\,km$ Given: $\frac{dx}{dt}=80\,km/h$ Differentiating $2^{2}+x^{2}=h^{2}$ with respect to $t$, we obtain $2x\frac{dx}{dt}=2h\frac{dh}{dt}$ $\implies \frac{dh}{dt}=\frac{x}{h}\frac{dx}{dt}$ $=\frac{6}{2\sqrt {10}}\times80\,km/h=24\sqrt {10}\,km/h\approx75.89\,km/h$