What is Rate of change: Definition and 279 Discussions

In mathematics, a rate is the ratio between two related quantities in different units. If the denominator of the ratio is expressed as a single unit of one of these quantities, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the numerator of the ratio expresses the corresponding rate of change in the other (dependent) variable.
One common type of rate is "per unit of time", such as speed, heart rate and flux. Ratios that have a non-time denominator include exchange rates, literacy rates, and electric field (in volts per meter).
In describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate (for example a heart rate is expressed "beats per minute"). A rate defined using two numbers of the same units (such as tax rates) or counts (such as literacy rate) will result in a dimensionless quantity, which can be expressed as a percentage (for example, the global literacy rate in 1998 was 80%), fraction, or multiple.
Often rate is a synonym of rhythm or frequency, a count per second (i.e., hertz); e.g., radio frequencies, heart rates, or sample rates.

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  1. T

    Finding the rate of change of the distance between 2 cars

    Homework Statement Car A is traveling north on Highway 16 and car B is traveling west on Highway 83. Each car is approaching the intersection of these highways. At a certain moment, car A is 0.3 km from the intersection and traveling at 90km/h while car B is 0.4km from the intersection and...
  2. T

    - Integration, Rate of Change, and Volume of Revolution Questions

    Please Help! - Integration, Rate of Change, and Volume of Revolution Questions Hi I have completes these following questions but am not sure if I have done them correctly as it is a long time since i studied these topics. I would really appreciate any help. :-) 1 Simplify the following as...
  3. S

    Calculating Rate of Change of Water in a Semi-Circular Bathtub

    Homework Statement A bathtub with shape semi-circle has radius 5 m is filled with water with rate 2 m^3/minute. How fast will the height of water in the bathtub change when it is 2 m from the bottom? Given that the volume of water when the radius = R and height = y is πy^2(R - y/3)...
  4. W

    What is the Instantaneous Rate of Change at X=2 for the Function F(x) = 4/(x-1)?

    Homework Statement Find the irc at X=2 F(x) = \frac{4}{x-1} Homework Equations \stackrel{lim}{h\rightarrow0}\frac{F(a+h)-F(a)}{h} The Attempt at a Solution I end up with \frac{4}{h+h^{2}} or some other form of an h on the bottom. Is there something else i can do to it??
  5. M

    Calculus - find average rate of change of the function over a given interval

    Calculus - please help find average rate of change of the function over a given interval Homework Statement h(t) = sin t, [3pi/4,4pi/3] please help me solve this, try to give me an explanation on every step please. i checked but i can't find anything on this in the book. i am...
  6. Maroc

    How to get instantaneous rate of change

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  7. T

    Rate of change of the magnetic field

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  8. Chewy0087

    Connected rate of change & Calculus

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  9. J

    How Does the Height of a Parachute Change with Horizontal Displacement?

    the height h(x) in meters above the ground of a parachute varies with her horizontal distplacement x in meters from a landing target on the ground as h(x) = 50sin^-1 (0.1x). what is the rate of change of h with respect to x at x = 6m? i was wondering of someone could help me with this question :)
  10. K

    Rate of change of acceleration?

    Do physicists ever have to account for the rate of change of acceleration? What about the rate of change of the rate of change of acceleration? This could keep going on forever, but is there some point where it just doesn't matter for a fairly accurate estimate?
  11. D

    Simple Precalculus: Average rate of change

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  12. M

    Time rate of change for electric flux help

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  13. A

    Instantaneous rate of change homework

    Homework Statement For the function f(x)=3x^2-1, for what x values is the instantaneous rate of change equal to the average rate of change on the interval [1,3] Homework Equations The Attempt at a Solution
  14. James R

    Rate of change of angle between a particle and a field

    I am wondering if anybody here can help me with a proof. A particle is moving with velocity \vec{v} in a field \vec{F} (which could be an electric or magnetic field, for example). The velocity can change with time as the particle moves, but the field is static in space. The particle sees the...
  15. Z

    Rate of change of radius through a a circular wire loop?

    A circular wire loop of radius r = 19 cm is immersed in a uniform magnetic field B = 0.690 T with its plane normal to the direction of the field. If the field magnitude then decreases at a constant rate of −1.0×10^-2 T/s, at what rate should r increase so that the induced emf within the loop is...
  16. C

    Average Rate of Change Using MVT for Derivatives

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  17. E

    Rate of Change: Distance Between Moving Point and Fixed Point

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  18. E

    Rate of change of water in a trough

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  19. S

    Shadow Rate of Change: Solving a Geometry Problem

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  20. U

    Finding the rate of change of an angle in triangle.

    Hello again! The base of an isosceles triangle is 20 cm and the altitude is increasing at the rate of 1 cm/min. At what rate is the base angle increasing when the area is 100 cm^2? I think I have found the solution, but want to have someone else check it over b/c I am not 100% sure...
  21. U

    Finding rate of change of shadow problem.

    Question: A 1.85 m tall man is walking toward a 12 m tall street light at night at a rate of 2.2 m/s. How fast is the length of his shadow changing when he is 12 m from the street light? so, using similar triangles, i got that (x+y)/12 = x/1.85. I can rearrange this into x= or y=, but then I...
  22. S

    Finding the Width of a Rectangle at a Changing Rate: A Related Rates Problem

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  23. P

    Rate of Change of the Angle of Elevation

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  24. L

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  25. A

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  26. R

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  27. A

    Rate of change problem very hard

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  28. J

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  29. B

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  30. F

    Calculating Rate of Change for Sphere Volume and Surface Area

    hi everyone, i am having trouble with this rate of change question. I need to find the rate of change of a sphere with a volume of V= 4/3*pi*r^3 in respect to it's surface area of A= 4*pi*r^2 can anyone give me a hand? Thanks
  31. A

    Rate of change of distance from origin?

    Homework Statement A particle moves along the parabola y = 4x^{2} such that it's x coordinate increases at a steady rate of 5 units/minute. How fast is the particle's distance from the origin changing when it is at the point (1,4)? Homework Equations The Attempt at a Solution...
  32. J

    Rate of Change Mass vs Time

    I have been reading a lot about rockets and how the mass flow rate is calculated. If i had a graph of the change in mass with respect to time of a rocket motor throughout its burn. How would i calculate the mass flow rate (change in mass at any given time t) ? Besides using density...
  33. A

    Differential equation questions, rate of change

    Hello and greetings everyone. 1. A quantity of oil is dropped into water. When the oil hits the water it spreads out as a circle. The radius of the circle is r cm after t seconds and when t = 3 the radius of the circle is increasing at the rate of 0.5 centimetres per second One observer...
  34. D

    Rate of change in distance question

    Okay, I'm 99% sure I've got the right answer here, but I just wanted to make certain before I send my assignment in. It's the last question and has been bugging me for the last few days until I had an eureka moment just a few minutes back. (In case you're wondering, I'm doing my studies by...
  35. I

    Water Tank Rate of Change Problem: Find the Rate at Which Water Level is Rising

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  36. I

    Rate of Change Problem: Finding Derivative of Temperature Function T(t)

    Homework Statement Suppose that t hours after a piece of food is put in the fridge its temperature (in Celsius) is T(t) = 15 - 3t + \frac{4}{t - 1} where 0 \leq t \leq 5. Find the rate of change of temperature after one hour. The Attempt at a Solution Since it's asking for...
  37. I

    Instantaneous rate of change of a sphere

    Homework Statement Find the instantaneous rate of change of V with respect to r when r=5 micrometers Homework Equations V=4/3pi(r^3) The Attempt at a Solution would you just take the derivative? and if so, wouldn't it just be zero, because it comes out to be a real number, right...
  38. I

    Calculating the Rate of Change of Area Covered on a Level Surface

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  39. C

    What Is the Rate of Change of Momentum When Water Hits a Building?

    Tried this problem, but not getting it. Homework Statement A fire hose sends 2528 gallons of water per minute against a building. The water strikes the building at 28 m/s and does not bounce back. What is the magnitude of the rate of change of momentum of the water?Homework Equations The...
  40. S

    Century Growth Problem: Rate of Change

    [SOLVED] Century Growth Problem: Rate of Change 1. Problem: Because Earth's rotation is gradually slowing, the length of each day increases: The day at the end of 1.0 century is 1.0 ms longer than the day at the start of the century. In 27 centuries, what is the total (in hours) of the daily...
  41. K

    Rate of Change - Just checking

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  42. P

    Finding Rate of Change and Time of Temperature Change in a Liquid System

    Homework Statement The temparature T of a liquid at a time t minutes is given by the equation T = 30 + 20e-0.05t, for t > 0 (1)Write down the initial tempatature o the liquid, and find the intial rate of change of temparature. (2)Find the time at which the temparature is 40...
  43. A

    A Rate of change and a differentials problem

    First off i'd like to say Hi to the forums, hehe. I didn't really see a new member area but I suppose this will do. Right now I'm kind of struggling with these two problems that I recently took a quiz on and didn't do so well. I've been trying to figure out how to work them out but i just don't...
  44. A

    Finding the Rate of Change of Shadow: A Man 6 ft Tall

    Homework Statement A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light, (A) at what rate is the tip of his shadow moving? (B) at what rate is the length of his shadow changing?Homework...
  45. J

    What units will the partial derivative of an angular equation give?

    Homework Statement If I have an equation that describes L in terms of some constants and an angle. e.g. L=C1*\sqrt{(C2-Sin[angle])}+C1*\sqrt{(C2-Cos[angle])} Then, if I take the partial derivative of the above wrt angle, then I would get the unit change in L for a unit change in angle...
  46. P

    Rate of Change in Water Depth of Cone-Shaped Tank

    The base of a cone-shaped tank is a circle of radius 5 feet, and the vertex of the cone is 12 below the base. The tank is filled to a depth of 7 feet, and water is flowing out of the tank at a rate of 3 cubic feet per minute. Find the rate of change of the depth of the water in the tank.
  47. J

    Magnitude of the rate of change of momentum of water

    Rate of change of momentum Homework Statement Hey everybody this is my first post and I really hope you guys can help me out. Ok so here's the problem which is divided into three parts: The first part consists of this: A fire hose sends 1464 gallons of water perminutre against a burning...
  48. X

    Average Rate of Change (word problem)

    I'm having trouble understanding this word problem. A construction worker drops a bolt while woring on a high-rise building 320m above the ground. After t seconds, the bolt has fallen a distance of s metres, where s(t) = 320-5t^2, o < t < 8 a. Find the average velocity during the first...
  49. F

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    Homework Statement s is the position of a particle along the s-axis, find an expression for the velocity and acceleration and determine when v=0 s=1-3t^2 Homework Equations v=ds/dt , a=dv/dt The Attempt at a Solution No idea where to begin, my book is not very clear. Do I just...
  50. A

    Rate of change of the magnitude of the displacement

    If the displacement was given by \overrightarrow x (t) ( i.e vector valued function ) . Then the velocity is \overrightarrow v = \frac {d \overrightarrow x}{dt} The speed is the magnitude of v . But .. What is the derivative of the magintude of the displacement \frac {d...
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