A team has been working to convert diesel-powered cars to run just as efficiently on used cooking oil! They want to compare the mileage and speed of their prototype with that of the diesel-powered car.
The prototype is 100 meters south of an intersection, while the diesel car is 100 meters east of the intersection. Both vehicles start moving at the same time. The prototype moves north, toward the intersection, and the diesel car moves east, away from the intersection. If the prototype is traveling at a velocity of 3 meters per second and the diesel car is traveling at 2 meters per second, what is the rate of change of the distance between the cars after four seconds? Round off your answer to two decimal places.
To find the rate of change of the distance between the prototype and the diesel car after four seconds, we first need to determine their positions at that time.
Let's start by considering the prototype's position after four seconds. We know that it is initially 100 meters south of the intersection and is moving north at a velocity of 3 meters per second. Therefore, in four seconds, it would have traveled a distance of 4 seconds multiplied by 3 meters per second, which is 12 meters north of its starting position.
Now let's determine the diesel car's position after four seconds. We are told it is initially 100 meters east of the intersection and is moving east at a velocity of 2 meters per second. Therefore, in four seconds, it would have traveled a distance of 4 seconds multiplied by 2 meters per second, which is 8 meters east of its starting position.
Now that we know the positions of both vehicles after four seconds, we can calculate the distance between them.
The prototype is now 12 meters north of its starting position, and the diesel car is 8 meters east of its starting position. To calculate the distance between them, we can use the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
In this case, the distance between the prototype and the diesel car after four seconds can be calculated as the square root of (12^2 + 8^2). This is approximately equal to the square root of 144 + 64, which is equal to the square root of 208.
To calculate the rate of change of this distance, we subtract the initial distance (100 meters) from the distance just calculated (approximately square root of 208). Therefore, the rate of change of the distance between the cars after four seconds is approximately square root of 208 minus 100.
To round off the answer to two decimal places, we simply round the square root of 208 minus 100 to the nearest hundredth.
Therefore, the rate of change of the distance between the prototype and the diesel car after four seconds is approximately equal to the square root of 208 minus 100, rounded off to two decimal places.