Solve the problem. Round your answer, if appropriate.
-One airplane is approaching an airport from the north at $182 \mathrm {~km} / \mathrm { hr }$. A second airplane approaches from the east at $294 \mathrm {~km} / \mathrm { hr }$. Find the rate at which the distance between the planes changes when the southbound plane is 31 $\mathrm { km }$ away from the airport and the westbound plane is $25 \mathrm {~km}$ from the airport.
A) $- 326 \mathrm {~km} / \mathrm { hr }$
B) $- 652 \mathrm {~km} / \mathrm { hr }$
C) $- 489 \mathrm {~km} / \mathrm { hr }$
D) $- 163 \mathrm {~km} / \mathrm { hr }$

Question

Quizplus · Accepted Answer

The rate at which the distance between the planes changes can be found using the Pythagorean theorem to first find the distance between the planes and then differentiating to find the rate of change. Given the speeds and distances from the airport, the distance between the planes is the hypotenuse of a right triangle with sides 31 km and 25 km. The rate of change of this distance with respect to time is the derivative of the distance with respect to time, which involves the speeds of the planes. Calculating directly, the combined speed (the rate at which the distance between the planes changes) can be found using vector addition of the velocity vectors of the two planes, which gives a result of 326 km/hr. This calculation considers the direction of the velocities, hence the negative sign is not applicable in the context of the magnitude of speed.

Solve the Problem 182 km/hr182 \mathrm {~km} / \mathrm { hr }182 km/hr

Solve the Problem $182 \mathrm {~km} / \mathrm { hr }$