Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Two cars leave an intersection at the same time. One travels north at 40 miles per hour, and the other travels east at 30 miles per hour. If $z$ represents the distance between the cars, (a) describe how the rate at which the cars are separating is related to the rates at which they are traveling. (b) At what rate is the distance between the two cars changing at the end of two hours?

(a) $\frac{d z}{d t}=\frac{x \frac{d x}{d t}+y \frac{d y}{d t}}{z}$(b) $50 \mathrm{mph}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 10

Related Rates

Derivatives

Missouri State University

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

06:34

(a) One car leaves a given…

04:34

a. One car leaves a given …

06:16

Car $A$ is driving east to…

05:55

Two cars approach an inter…

04:26

Car $A$ is driving south, …

03:13

Two cars start moving from…

this problem. We're looking at two cars leaving an intersection. Now, Anytime you have a related rate problem, it is best to draw a picture. If you can't, it makes it much easier to visualize how all of our numbers and letters relate to each other. So here's our intersection. We have one car that's going north and we have one car that's going east. And let's call these distances X and why. Okay, so what are we told? We're told that the one going north is traveling at 40 MPH. Well, speed is a change that's telling me how fast D Y is changing about. Sorry, how fast? Why is changing with respect to time So D Y d t is the speed of that north bound car or 40 MPH. Likewise, we're told that one going east is going 30 MPH, so that's how fast Distance X is changing. That is 30 MPH. Usually, our rates are in regarding some unit of time, so MPH inches per second, gallons per month. Whatever it is, it's usually per some unit of time, so MPH. We know we have rates d y d t d x d t and we're going to do two parts to this problem. But first we want to describe how the rate at which the car's air separating is related to the rates that they're traveling. So we're going to add one more piece to this. We're told that a distance Z is the distance between our two cars, and we want to know how fast Z is changing. So what we're looking for is DZ d T. That's our question. And we want to relate that to, um the distances are the rates at which both of these cars are traveling. So we're gonna have relate this to y and X d Y DT dx DT. So when we do a related rate, we need to have some formula or equation or function that ties my variables together. In this case, we have a right triangle, so that's gonna be the Pythagorean serum Z squared equals X squared plus y squared. Now let's take our derivative. We can use implicit differentiation here. We're doing everything with respect to time. So on the left here I have to z and then DC DT I have two x dx DT plus two y d Y D t. Every single term has a two, so I can cancel. And now I saw for DC DT because that's what we're looking for. So that gives me X times, dx, DT plus why times D Y. D t divided by c So that is how fast that distance between the two cars is changed. Now let's apply this. At what rate is the distance between the two cars changing at the end of two hours? So now, doing a certain time I want to see to be two hours. Well, what do we know after two hours? I know how far the North bound car has gone in two hours. It's going 40 MPH. In two hours, it will have gone 80 miles. What about the car going east? While it's going 30 MPH in two hours, it's gone 60 miles, and I can find out how fast or how far Z is at that particular point. At two hours, I use Pythagorean theory. Um, Z is going to be the square root of X squared plus why squared and when you do all that out, You get Z equaling 100. So we have everything we need to solve for D C D t my ex again. We didn't try to substitute or find numbers to find our rate. That's just our generic. Now we have our position, a specific point in time. We know all of our variables. We can plug things in exit. This particular point in time is 60 dx DT. That's my speed. That's 30 plus Why? Why is 80 d Y? D? T is 40 MPH over Z, which is 100 do this all out and simplify. We end up with 50 MPH. That's how fast Dizzy D T is changing at that point.

View More Answers From This Book

Find Another Textbook

01:57

A 12 inch piece of wire is to be cut into two pieces. One piece is to be use…

02:01

A house and lot valued at $100,000$ is being depreciated over 25 years by th…

02:36

Find $d y / d x$ using any method.$$x y=7$$

01:52

Find all the asymptotes and plot the graph of the given function.$$g(x)=…

01:43

Locate all critical points.$$h(x)=4 x^{3}-13 x^{2}+12 x+9$$

01:12

Use your knowledge of the derivative to compute the limit given.$$\lim _…

01:30

Let $f(x)=x^{-1 / 2}$ on the interval (0,1] . (a) Does $f$ satisfy the condi…

01:22

Determine where the function is concave upward and downward, and list all in…

03:45

Linearize $f$ near $x_{0}$.$f(x)=3 x^{2}-2 x+5$(a) $x_{0}=0$(b) …

05:24

Determine the equation of the tangent line to the given curve at the indicat…