Solve each problem. At 9: 00 A.M., Car A is traveling north...

Solve each problem. At 9: 00 A.M., Car A is traveling north at 50 mph and is located 50 miles south of Car $B$. Car B is traveling west at 20 mph. (a) Let (0,0) be the initial coordinates of Car B in the xy-plane, where units are in miles. Plot the locations of each car at 9: 00 A.M. and at 11: 00 A.M. (b) Find the distance between the cars at 11: 00 A.M.

Solve each problem.
At 9: 00 A.M., Car A is traveling north at 50 mph and is located 50 miles south of Car $B$. Car B is traveling west at 20 mph.
(a) Let (0,0) be the initial coordinates of Car B in the xy-plane, where units are in miles. Plot the locations of each car at 9: 00 A.M. and at 11: 00 A.M.
(b) Find the distance between the cars at 11: 00 A.M.

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Key Concepts

Coordinate System

A coordinate system, often represented as an xy-plane, is used to locate points in two-dimensional space. It provides a frame of reference by establishing axes (typically horizontal and vertical) where positions are denoted by ordered pairs, making it easier to analyze the locations and movements of objects in problems like vehicle motion.

Parametric Representation of Motion

Parametric equations are used to express the coordinates of moving objects as functions of time. In problems involving constant speed and direction, these equations allow one to separately track the x and y components of each object's position, simplifying the analysis of their trajectories and helping to determine their locations at any given time.

Distance Formula

The distance formula, derived from the Pythagorean theorem, is used to calculate the straight-line distance between two points in the plane. By computing the differences in the corresponding x and y coordinates and applying the square root of the sum of their squares, one can accurately determine the separation between objects at a specific time.

Transcript

00:01 So it is given to me in this question that at 9 a .m.

00:03 A car is traveling north at 50 miles per hour.

00:07 So let me name it car a.

00:10 Speed of car a is given to me as 50 miles per hour.

00:14 And speed of car b is given to me 20 miles per hour.

00:20 So a is traveling northwards and b is traveling westwards.

00:28 So i have this coordinate axis system over here.

00:33 If i call this the x -axis and this is the y -axis.

00:36 So this direction represents east.

00:40 This represents west and this will represent the north side.

00:45 So let 0 -0 be the initial coordinates of kar -b in the xy plane.

00:52 So at 9 o 'clock when they are talking, this is the point where carby is positioned.

00:57 Now car b will be traveling on the west side with 20 miles per hour and it is given that car a is located 50 miles south of car b so car a will be somewhere over here.

01:15 It will be traveling upwards and this distance is given to me as 50 miles.

01:23 So the first question is plot the location of each car at 9 a .m.

01:29 And 11.

01:29 So this is the position of a and b.

01:34 I can name this coordinates as 0 comma minus 50.

01:39 All right...

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