Answer to Question #177540 in Calculus for kienth bryan doroin

Question

Starting from the same point, Biden starts walking eastward at 52 m/s while Trump starts running towards the south at 86 m/s. How fast is the distance between Biden and Trump increasing after 2 seconds?

Answer 1

$let\ x\ represent\ Biden's\ distance\ from\ the\ origin\ point$

$\ \ \ \ \ \ y\ represent\ Trump's\ distance\ from\ the\ origin\ point$

$\ \ \ \ \ \ t\ represent\ the\ time\ after\ they\ started\ walking$

$\ \ \ \ \ z\ represent\ the\ distance\ between\ them$

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

$\frac{dx}{dt}=52$

$\frac{dy}{dt}=86$

$we\ wish\ to\ calculate\frac{dz}{dt}$

$\int{\frac{dx}{dt}dt}=52t+c$

$\int{\frac{dy}{dt}dt}=86t+k$

$at\ t=0,\$

$x=0,\ y=0$

$52\left(0\right)+c=0\rightarrow c=0\rightarrow x=52t$

$86\left(0\right)+k=0\rightarrow k=0\rightarrow y=86t$

$z=\sqrt{x^2+y^2}=\ \sqrt{\left(52t\right)^2+\left(86t\right)^2}=\ \sqrt{10100}\ t$

$\frac{dz}{dt}=\ \sqrt{10100}=100.499m/s$