Solve the problem. Round your answer, if appropriate.
-One airplane is approaching an airport from the north at 151 km/hr. A second airplane approaches from the east at 203 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is 26 km away from the airport and the westbound plane is 15 km from the airport.

Question

Quizplus · Accepted Answer

The rate at which the distance between the planes changes can be found using the Pythagorean theorem in a right triangle, where the legs represent the distances of the planes from the airport and the hypotenuse represents the distance between the planes. Differentiating with respect to time gives us the rate of change of the distance between the planes. Given the distances (26 km and 15 km) and the speeds (151 km/hr and 203 km/hr), we can calculate this rate. The calculation is as follows: Let $x$ be the distance of the northbound plane from the airport, and $y$ be the distance of the eastbound plane from the airport. The distance between the planes, $z$, can be found using the Pythagorean theorem: $z^2 = x^2 + y^2$.Differentiating both sides with respect to $t$ gives: $2z \frac{dz}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt}$.Given that $\frac{dx}{dt} = -151$ km/hr (since the northbound plane is approaching the airport, the distance $x$ is decreasing), and $\frac{dy}{dt} = -203$ km/hr (since the eastbound plane is also approaching, the distance $y$ is decreasing), and at the instant when $x = 26$ km and $y = 15$ km, we find $z$ using the Pythagorean theorem: $z = \sqrt{26^2 + 15^2} = \sqrt{676 + 225} = \sqrt{901}$.Plugging the values into the differentiated equation: $2\sqrt{901} \frac{dz}{dt} = 2(26)(-151) + 2(15)(-203)$.Solving for $\frac{dz}{dt}$ gives: $\frac{dz}{dt} = \frac{-2(26)(151) - 2(15)(203)}{2\sqrt{901}}$.This simplifies to: $\frac{dz}{dt} = \frac{-7826}{2\sqrt{901}} = \frac{-7826}{2(30.0166)} \approx -232$ km/hr.Therefore, the rate at which the distance between the planes changes is approximately -232 km/hr, indicating they are getting closer at this rate.

Solve the Problem. Round Your Answer, If Appropriate