Two cars approach an intersection, one heading east at the rate of 30 km / h and the other head

step 1 We have two cars approaching an intersection. So let's say this is our intersection. One is head

Two cars approach an intersection, one heading east at the...

Two cars approach an intersection, one heading east at the rate of $30 \mathrm{~km} / \mathrm{h}$ and the other heading south at the rate of $40 \mathrm{~km} / \mathrm{h}$. At what rate are the two cars approaching each other at the instant when the first car is $100 \mathrm{~m}$ from the intersection and the second car is $75 \mathrm{~m}$ from the intersection? Assume the cars maintain their respective speeds.

Key Concepts

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to time by relating it to other quantities whose rates of change are known. In these problems, an equation establishes a relationship among several variables, and by differentiating that equation with respect to time, one can solve for the unknown rate of change.

Pythagorean Theorem

The Pythagorean Theorem provides a relationship between the sides of a right triangle. In a variety of related rates problems, this theorem is used to relate distances that form perpendicular components, allowing us to express the distance between two moving objects as the hypotenuse of a right triangle.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the variables are interdependent, without explicitly solving one variable in terms of another. This technique is crucial in related rates problems, as it allows the differentiation of relationships between variables with respect to time, facilitating the determination of how one rate of change affects another.

Transcript

00:01 We have two cars approaching an intersection.

00:03 So let's say this is our intersection.

00:07 One is heading east, so in this direction, at a rate of 30 kilometers per hour.

00:16 The other car is heading south, so here, at 40 kilometers per hour.

00:26 So at what rate are the two cars approaching each other when the first car is 100 meters from the intersection and the second car is 75 meters? okay, so let's say the car is right over here and the distance is 100 meters and the other car is here and the distance is 75.

00:54 Okay, so we'll also label this x and we'll label that y.

00:58 So the distance from each other would be the hypotenuse here.

01:06 So we'll just call that capital b.

01:10 Okay, so let's write out what we have first.

01:13 We have dx dt is 30 kilometers an hour, but it's actually negative because this distance here is decreasing.

01:27 Okay, so the only other thing is notice that the speed is given in kilometers, but this distance here is actually given in meters.

01:39 Okay, so you need to make sure that you pick one and stick with it.

01:42 So let's choose kilometers.

01:45 So in this case, d xdt would be negative 30, and this is kilometers per hour.

01:52 And then d ydt would also be negative because this distance is decreasing.

01:58 So this is negative 40 kilometers per hour.

02:04 Okay, so now what we want is we want to find at what rate the two cars are approaching each other.

02:11 So we want to find d, d, d, t, and this is evaluated at so we'll call that xy.

02:20 And this is equal to.

02:21 So 100 meters, that would be 0 .1 kilometers...

Please give Ace some feedback