What is Related rates: Definition and 371 Discussions

In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time. Because science and engineering often relate quantities to each other, the methods of related rates have broad applications in these fields. Differentiation with respect to time or one of the other variables requires application of the chain rule, since most problems involve several variables.
Fundamentally, if a function



F


{\displaystyle F}
is defined such that



F
=
f
(
x
)


{\displaystyle F=f(x)}
, then the derivative of the function



F


{\displaystyle F}
can be taken with respect to another variable. We assume



x


{\displaystyle x}
is a function of



t


{\displaystyle t}
, i.e.



x
=
g
(
t
)


{\displaystyle x=g(t)}
. Then



F
=
f
(
g
(
t
)
)


{\displaystyle F=f(g(t))}
, so





F


=

f


(
g
(
t
)
)


g


(
t
)


{\displaystyle F'=f'(g(t))\cdot g'(t)}
Written in Leibniz notation, this is:







d
F


d
t



=



d
f


d
x







d
x


d
t



.


{\displaystyle {\frac {dF}{dt}}={\frac {df}{dx}}\cdot {\frac {dx}{dt}}.}
Thus, if it is known how



x


{\displaystyle x}
changes with respect to



t


{\displaystyle t}
, then we can determine how



F


{\displaystyle F}
changes with respect to



t


{\displaystyle t}
and vice versa. We can extend this application of the chain rule with the sum, difference, product and quotient rules of calculus, etc.
For example, if



F
(
x
)
=
G
(
y
)
+
H
(
z
)


{\displaystyle F(x)=G(y)+H(z)}
then







d
F


d
x







d
x


d
t



=



d
G


d
y







d
y


d
t



+



d
H


d
z







d
z


d
t



.


{\displaystyle {\frac {dF}{dx}}\cdot {\frac {dx}{dt}}={\frac {dG}{dy}}\cdot {\frac {dy}{dt}}+{\frac {dH}{dz}}\cdot {\frac {dz}{dt}}.}

View More On Wikipedia.org
  1. karush

    Related rates: How fast is a runner going to 1st base, as seen from 2nd base?

    TL;DR Summary: find how fast the distance is changing from 3rd base to a runer going to first base Mentor note: Thread has been moved from a technical math section, so is missing the homework template. A baseball diamond is a square with side 90 feet. A batter hits the ball and runs toward...
  2. N

    B Infinitely smaller changes in a related rates question

    I was looking at related rates problems, and the problem of finding the rate at which the area of a circle changes with respect to time. In order for the area of a circle to be changing at a constant rate, say per second, it would have to mean that the radius was increasing by a smaller and...
  3. rxh140630

    Related rates: water level in a cone

    Firstly I'm having trouble understanding what water level means. I tried a quick google search and got the following: " water level(Noun) The level of a body of water, especially when measured above a datum line. " That doesn't help me. Is water level the distance from the base to where the...
  4. karush

    MHB Apc.2.8.1 ap vertical circular cylinder related rates

    $\tiny{2.8.1}$ The vertical circular cylinder has radius r ft and height h ft. If the height and radius both increase at the constant rate of 2 ft/sec, Then what is the rate at which the lateral surface area increases? \een $\begin{array}{ll} a&4\pi r\\ b&2\pi(r+h)\\ c&4\pi(r+h)\\ d&4\pi rh\\...
  5. archaic

    Seemingly a related rates problem

    The way I imagine it is that the two uncharged balls will move only vertically, while the charged balls will be coming down, and getting away from each other horizontally. If I focus on just one side, then I notice that ##L## is decreasing from above and below by the same amount (same vertical...
  6. jaychay

    MHB Calculus airplane related rates problem ( cosine rule)

    A student has test his airplane and he is far from the airplane for 5 meter.He start to test his airplane by letting his airplane to move 60 degree from the horizontal plane with constant velocity for 120 meter per minute.Find the rate of distance between the student and the plane when the plane...
  7. jaychay

    Related rates calculus problem about a water tank

    Summary:: Consider the rectangular water tank, at the base the length is the same for 200 cm. There are 100 holes for water to come out which each hole have the same flow rate. Find the amount of water that come out in each hole by using differential when we know that there is an error in the...
  8. jaychay

    MHB Related rates calculus problem about water tank

    Consider the rectangular water tank, at the base the length is the same for 200 cm. There are 100 holes for water to come out which each hole have the same flow rate. Find the amount of water that come out in each hole by using differential when we know that there is an error in the measurement...
  9. karush

    MHB Why is the e in the derivative for related rates?

    Water is pumped into a tank at a rate of $r(t) = 30(1-e^{e-0.16t})$ gallons per minute, where t is the number of minutes since the pump was turned on. If the tank contained 800 gallons of water when the pump was turned on, how much water, to the nearest gallon, is in the tank after 20 minutes...
  10. karush

    MHB 2.6.284 AP calculus Exam Lamp and Shadow related rates

    yes I know this is a very common problem but likewise many ways to solve it ok I really have a hard time with these took me 2 hours to do this looked at some examples but some had 3 variables and 10 steps confusing to get the ratios set up... ok my take on it is here see if you can solve...
  11. JosephTraverso2

    Find the derivative of a balloon's circumference

    A balloon's volume is increasing at a rate of dV/dt. Express the rate of change of the circumference with respect to time (dc/dt) in terms of the volume and radius. Homework Equations Vsphere = (4/3)(π)(r^3) C = (2)(π)(r) The Attempt at a Solution [/B] My strategy was to come up with two...
  12. Michael Santos

    Related rates & unknown factors

    Homework Statement Your blowing up a balloon at a rate of 300 cubic inches per minute. When the balloon's radius is 3 inches, how fast is the radius increasing? Homework EquationsThe Attempt at a Solution I know the answer to this question. It is approximately 2.65 inches per minute, what my...
  13. opus

    A different related rates problem - no time mentioned

    Homework Statement Evening all. I have a related rates problem that I haven't come across which doesn't seem to involve time which is usually the independent variable that we take the derivative with respect to. It has thrown me for a loop. Find the change in the volume of a right cylinder...
  14. opus

    Differentiation with units- Related Rates problem

    Homework Statement A spherical balloon is being filled with air at a constant rate of ##2cm^3/sec##. How fast is the radius increasing when the radius is 3 cm? Homework EquationsThe Attempt at a Solution This is a problem given and spelled out in my text. However, it seems to be assuming that...
  15. karush

    MHB 214 related rates gravel cone

    Gravel is being dumped from a conveyor belt at a rate of $30\displaystyle\frac{ ft}{min}$ and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is $10 ft$ high...
  16. T

    MHB Am I doing this related rates right?

    Hey guys, I have this related rates problem that I'm working through. I think I might have an answer, but I'm not sure. Here's the question. A potter shapes a lump of clay into a cylinder using a pottery wheel. As it spins, it becomes taller and thinner, so the height, h, is increas- ing and...
  17. L

    Related rates problem - kinda stuck on this one

    Homework Statement Suppose you take a car trip, traveling east along a very long highway, starting at time t 0. Let x t be the number of gallons of gasoline used during the first t hours, and let st be the distance traveled in that time. Because you’re using very cheap gasoline that’s not good...
  18. starstruck_

    Related rates (prism with a trapezoid base)

    Homework Statement A trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has a height of 50 cm. If the trough is being filled with water at the rate of 0.2m3/min, how fast is the water level rising when the...
  19. starstruck_

    Finding the Rate of Change of Water Height in a Filling Cylindrical Tank

    Homework Statement ((I cannot, for the love of life, understand related rates, so please bear with me. Thank you! )) A cylindrical tank with radius 5m is being filled with water at a rate of 3m3/min. How fast is the height of the water increasing? I'm having trouble interpreting this...
  20. karush

    MHB 2214.16 Related Rates With Draining Cone

    Water is being drained from a container which has the shape of an inverted right circular cone The container has a radius of $6 in$ at the top and a length of $8 in$ at the bottom. when the water in the container is $6 in$ deep, the surface level is falling at a rate of $0.9 in/sec$ find the...
  21. K

    Question: Shadow Speed Related Rates

    Homework Statement 2. Homework Equations Similar triangles The Attempt at a Solution $$\frac{y}{30}=\frac{50}{x}~\rightarrow~x=\frac{1500}{y}$$ $$\frac{dx}{dt}=-\frac{1500}{y^2}$$ $$s=16t^2=16\frac{1}{4}=4$$ $$\frac{dx}{dt}=-\frac{1500}{16}$$ The answer should be ##~\displaystyle...
  22. A

    What is the rate of change of the area of a rectangle?

    Homework Statement Find the rate of change of the area of a rectangle whose area is 75cm^2. The length is 3 times the width. The rate of change of the width is 2cm/second. Homework EquationsThe Attempt at a Solution A=75 A'=? L=3x= 15 L'=6 W=x=5 W'=2 A'=L'W+LW' A'= (6)(5)+ (15)(2) A'=60cm/sec
  23. Schaus

    Related Rates stone drop

    Homework Statement A stone is dropped into some water and a circle of radius r is formed and slowly expands. The perimeter of the circle is increasing at 3 m/s. At the moment the radius is exactly 2m, what rate is the radius of the circle increasing? Answer ## \frac {dr} {dt} = 0.48 m/s##...
  24. harpazo

    MHB How to Solve Related Rates Problems in Calculus 1

    I decided to review a few calculus 1 topics of interest. I like related rates but setting up the proper equation has been a big problem for me. Question: A light is on top of a building that is 15 ft. high. A man, 6 ft. tall, is walking away from the building at the rate of 2 ft/sec. At what...
  25. H

    2 Related Rates Questions (Calc I)

    1. Suppose we have ladder laying against a wall, with x sub 0 = 1 and y sub 0 = 1. Given that y(1) = 1 and x′(1) = 1, find y'(1). Okay so using Pythagorean's Theorem x^2 + y^2 = L, I found that the remaining side is sqrt(2). After taking the derivative I got 2x(t)x'(t) + 2y(t)y'(t) = 0. The...
  26. T

    MHB Related rates and differential

    Hi, need some help on the following question. Just want to check on part a on the followingv=4/3\pi.r^3 dv = 4\pi.r^2 dr dv/dt = 4\pi.r^2 dr/dt dr/dt = (dv/dt)/ 4\pi.r^2 dr/dt = (-KA)/4\pi.r^2 dr/dt= -K part B need some help Thanks Tom
  27. A

    MHB Need help on related rates problem

    I'm having problems with these related rates problems. Can you please show me how you solve the problem along with an explanation. Here is the problem: 1. A water tank has the shape of an inverted right (non oblique) cone with a diameter of 20 meters and a depth of 14 meters. Water is flowing...
  28. A

    Related rates of a ladder sliding

    Homework Statement A 5 m ladder is sliding down the wall, and h is the height of the ladder's top at time t, and x is the distance from the wall to the ladder's bottom at time t. Given that h(0) = 4 at t = 0 seconds and dh/dt = 1.2m/s, and the ladder is 5m long find x(2) and dx/dt at t=2...
  29. C

    Related rates - finding hypotenuse of triangle

    Homework Statement A plane flying horizontally at an altitude of 3 mi and a speed of 480 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 4 mi away from the station. (Round your answer to the nearest whole...
  30. karush

    MHB 205.23 related rates sphere volume

    $\tiny {205.23} $ $\text{ The volume }\displaystyle V=\frac{4}{3}\pi{r}^{3} \text{ of a spherical balloon changes with the radius.} $ $\text{a) at what rate } \displaystyle \frac{ft^3}{ft} \\$ $\text{does the volume change with respect to the radius when $r=2 ft.$} $...
  31. karush

    MHB Rate of Change in Area of Triangle with Fixed Sides

    $29.$ Two sides of a triangle are $4 \, m$ and $5 \, m$ in length and the angle between them is increasing at a rate of $\frac{0.06 \, rad}{s}$ Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length. $\frac{\pi}{3}$ $\displaystyle...
  32. Alexanddros81

    Related Rates of Accelerations in Pulley Problem

    Homework Statement Hi! I am new to this forum and i have problem understanding sample problem 13.5 from Vector Mechanics for Engineers 10th edition Statics and dynamics. (Beer) In particular I don't understant in part b how he comes to the assumption aC=-1/2aD If anyone can help thanks...
  33. MrDickinson

    B I with a related rates question

    A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. Two ratios are proportional if they change equally and are related by a constant of proportionality? Not sure about this definition, but please correct it if you can...
  34. MrDickinson

    Need a little help with this related rates problem

    Can someone help me with this? (dA/dt)=1cm/s (cm^2 whatever...leave out trivial corrections). A=pir^2 (dA/dt)=2pir(dR/dt) Multiply through by (1/2pir) (dA/dt)/(2pir)=dR/Dt What is the rate of change of the radius for a circumfrance of 2 I just used the related rates formula that I derived for...
  35. RobertColman

    How can I derive a formula for change in B field strength?

    I'm a bit stuck on this question (which is homework so hints are more welcome than outright answers). The question is: A very long wire carrying a current I is moving with speed v towards a small circular wire loop of radius r. The long wire is in the plane of the loop and is too long to be...
  36. 5

    Related rates problem help

    Homework Statement A 1.9 metre ladder is leaning against a vertical wall. If the bottom of the ladder is 30 cm from the wall and is being pull away from the wall with a horizontal speed of 25 cm per second, how fast is the top of the ladder sliding down the vertical wall? We define the the...
  37. C

    Related Rates Homework Practice: Examples and Solutions"

    Homework Statement I am having some trouble with related rates and would like some confirmation if I am on the right track and approaching the questions correctly. I have attached a few related rates problems that I have worked through below. Homework Equations N/A The Attempt at a...
  38. NatFex

    I Related Rates (+Trig) Question

    I'm a bit iffy with the whole of the 'related rates' topic of my calculus course. I've tried coming up with a question of my own to see if I can solve it. The question is as follows: The distance between a point on the ground and the bottom of a pole is 26m. The angle of inclination from that...
  39. R

    Calculus - Related Rates Problem

    Question: Two bikers leave a diner at the same time. Biker Slim rides at 85kmh [N] and Biker Haug rides at 120kmh [NE]. How fast is the distance between them changing 40 minutes after they left? I suggest looking at my photos of the triangles and such, as explaining it over text can be a bit...
  40. R

    MHB Help on Related Rates implicit differentiation

    Hi! I recently came upon this problem : the height of a right angled triangle is increasing at a rate of 5cm/min while the area is constant. How fast must the base be decreasing at the moment when the height is 5 times the base? I drew a picture of the triangle, labelled the height (h) and...
  41. Mr Davis 97

    Related Rates Sand Pile Problem

    Homework Statement A machine starts dumping sand at the rate of 20 m3/min, forming a pile in the shape of a cone. The height of the pile is always twice the length of the base diameter. After 5 minutes, how fast is the height increasing? After 5 minutes, how fast is the area of the base...
  42. M

    Linear & Angular Velocity Related Rates

    Homework Statement A rotating beacon is located 1 kilometer off a straight shoreline (see figure). If the beacon rotates at a rate of 3 revolutions per minute, how fast (in kilometers per hour) does the beam of light appear to be moving to a viewer who is 1/2 kilometer down the shoreline...
  43. M

    Related Rates: Volume of Cone

    Homework Statement A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep. Homework EquationsThe Attempt at a Solution...
  44. M

    Related Rates of Volume Change for Expanding Cube Edges

    Homework Statement All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter and (b) 10 centimeters? Homework EquationsThe Attempt at a Solution I used the equation for the volume of a cube: V = s3 but I'm not...
  45. T

    Time dilation/related rates

    1. A spacecraft going at .99c is heading straight towards a star that's at a distance of 60,000 light years. Another ship 25,000 light years below the first one also is heading towards the star also at .99c. What what is the related rate between the time dilation of the first spacecraft to...
  46. S

    Spherical balloon related rates problem

    Homework Statement You are blowing air into a balloon at a rate of 4*pi/3 cubic inches per second. (The reason for this strange-looking rate is that it will simplify your algebra a little bit.) Assume the radius of your balloon is zero at time zero. Let r(t), A(t) and V(t) denote the radius...
  47. N

    Related rates question involving volume of cone

    Homework Statement Sand falls from a conveyor belt at the rate of 10m^3/min onto the top of a conical pile. The height of the pile is always 3/8ths of the base diameter. How fast is the radius changing when the pile is 4 m high? 3. The Attempt at a Solution V = pir^2 (4/3) -- volume of a...
  48. T

    MHB Related Rates for a triangle

    Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is $$\pi/3$$ When the angle is $$pi/3$$, the third side is equal to...
Back
Top