An air traffic controller spots two airplanes at the same altitude converging to a point as they fly at right angles to each other. One airplane is 70 miles from the point and has a speed of 280 miles per hour. The other is 240 miles from the point and has a speed of 960 miles per hour. (a) At what rate is the distance between the planes changing

DL/dt = 1000 miles/hour

Step-by-step explanation:

Let´s call A Point for the airplane at 70 miles from the point O (converging point), and point B for the airplane at 240 miles from point O.

Then the three-points A, B, and O shape a right triangle with legs (distances from each of the airplane to point O, and hypotenuse L distance between the two airplanes

Then according to Pithagoras´theorem:

L²  =  (AO)²  + (BO)²

At the moment t when the airplanes are far away as 70 and 240 miles per hour

L²  = (70)²  + ( 240)²

L²  = 4900 + 57600

L = √62500

L = 250 miles

In general

L²  = x² + y²

That equation is always valid for a right triangle if the airplanes are approaching keeping the right triangle shape then:

L² = x² + y²        where x and y are the legs ( that legs change in time then):

Tacking derivatives on both sides of the equation

2*L*DL/dt = 2*x*Dx/dt + 2*y*Dy/dt

By substitution:    since Dx/dt = 280 m/h   and Dy/dt = 960 m/h

2*(250)*DL/dt = 2*70*280 + 2*(240)*960

500*DL/dt  =  39200 + 460800

DL/dt  =  500000/ 500

DL/dt = 1000 miles/hour

answer: i believe it’s 245.04

step-by-step explanation:

answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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