Related Rates IV

Problem 166: Two airplanes are flying in the air at the same height: airplane $A$ is flying east at 250 miles per hour and airplane $B$ is flying north at 300 miles per hour. If they are both heading to the same airport, located 30 miles east of airplane $A$ and 40 miles north of airplane $B$ , at what rate is the distance between the airplanes changing?

Solution: The two airplanes make a right triangle where the length of the two sides are given: $y = 30$ and $x =40$ . Finding the hypotenuse

(1) $\begin{equation*} z^2 & = x^2 + y^2 \quad \Rightarrow \quad z =\sqrt{20^2 + 40^2} = 50. \end{equation*}$

We need to find $dz/dt$ . Then

(2) $\begin{align*} z^2 & = y^2 + x^2 \\ 2z \frac{dz}{dt} & = 2x \frac{dx}{dt} + 2y\frac{dy}{dt} \\ 2(50) & = 2(30)(-250) + 2(40)(-300) \\ \frac{dz}{dt} & = \frac{-15000 - 24000}{100} \\ \frac{dz}{dt} & = -390 \ \text{mile}/\text{hrs}.\end{align*}$

Note that $dz/dt$ is negative because the distance between them is decreasing. Hence, the distance between the airplanes is changing at a rate of 390 miles/hrs.

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