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 cars, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) 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 forms a right-angled triangle, with one side measuring 100 meters (distance north for the prototype) and the other side measuring 100 meters (distance east for the diesel car).
Using the Pythagorean theorem, we can calculate the initial distance between the cars:
Distance^2 = (100^2) + (100^2)
Distance = √(10000 + 10000)
Distance = √20000
Distance ≈ 141.42 meters
Now, we can calculate the rate of change of the distance between the cars after four seconds. Since the prototype moves north at a velocity of 3 meters per second and the diesel car moves east at a velocity of 2 meters per second, we can think of their movements as vectors.
After four seconds, the prototype would have traveled a distance of 3 meters/second * 4 seconds = 12 meters north from its initial position.
Similarly, the diesel car would have traveled a distance of 2 meters/second * 4 seconds = 8 meters east from its initial position.
Now, we can use vector addition to find the new position of the prototype from the intersection:
Prototype position = (100 meters, -12 meters) from the intersection
Similarly, we can find the new position of the diesel car from the intersection:
Diesel car position = (100 meters, 8 meters) from the intersection
Now, we can calculate the new distance between the cars:
Distance^2 = (100 + 100 + 12)^2 + (-12 + 8)^2
Distance = √(212^2 + (-4)^2)
Distance = √44944 + 16
Distance = √44960
Distance ≈ 211.97 meters
The rate of change of the distance between the cars after four seconds would be the difference between the initial distance and the new distance:
Rate of change = Distance (new) - Distance (initial)
Rate of change = 211.97 - 141.42
Rate of change ≈ 70.55 meters
Rounded off to two decimal places, the rate of change of the distance between the cars after four seconds is approximately 70.55 meters.
To find the rate of change of the distance between the prototype and the diesel car after four seconds, we need to determine their positions at that time.
Let's start by calculating the prototype's position after four seconds. Since it is moving north at a velocity of 3 meters per second, the displacement after four seconds can be found by multiplying the velocity by the time: 3 meters/second * 4 seconds = 12 meters north.
Next, let's calculate the diesel car's position after four seconds. It is moving east at a velocity of 2 meters per second, so its displacement after four seconds is 2 meters/second * 4 seconds = 8 meters east.
Now we can find the distance between the two vehicles after four seconds. We can use the Pythagorean theorem for this. The horizontal distance (east-west) between the two vehicles is 100 meters (the initial distance), minus 8 meters (the displacement of the diesel car), which gives us 92 meters. The vertical distance (north-south) between the two vehicles is 12 meters (the displacement of the prototype).
Using the Pythagorean theorem: distance^2 = horizontal distance^2 + vertical distance^2, we can calculate the distance between the two vehicles after four seconds:
distance^2 = 92^2 + 12^2
distance^2 = 8464 + 144
distance^2 = 8608
distance ≈ √8608
distance ≈ 92.83 meters
Finally, we can calculate the rate of change of the distance between the two vehicles by dividing the change in distance (0 meters - 92.83 meters) by the change in time (4 seconds): rate of change = (0 meters - 92.83 meters) / 4 seconds.
rate of change ≈ -92.83 meters / 4 seconds
rate of change ≈ -23.21 meters/second
Therefore, the rate of change of the distance between the prototype and the diesel car after four seconds is approximately -23.21 meters/second.
The Y-coordinate of car 1, relative to the intersection, are
Y = -100 + 3t (Its X value is 0)
The X-coordinate of car 2, relative to the intersection, is
X = 100 + 2t (Its Y value is zero)
The distance between the two cars is
R = sqrt(X^2 + Y^2) = sqrt(2*10^4 -600t + 9t^2 +400t +4t^2) = sqrt(2*10^4 +13t^2 -200t)
The rate of change of separation distance is
dR/dt = (1/2)*(26t -200)/sqrt(2*10^4 +13t^2 -200t)
Compute the value of dR/dt at t = 4s.
(The denominator is the distance apart, which is 139.13 m at t = 4 s)