What are the steps to a related rate problem? What are the key words you look for and what do they mean? Give an example of a related rate problem and solve it.
What are the steps to a related rate problem? There are four major steps that can be used as a general guideline to solving a related rate problem however, it needs to be kept in mind that although there are set steps for working related rate problems, each problem is unique in its requirements for solving the problem completely. The steps to solving these problems are as follows:
First, identify all given quantities and quantities to be determined. Also make a sketch and be sure to label the quantities.
Second, you must choose and write an equation involving the variables whose rates of change either are given or are to be determined
Third, you must use the chain rule implicitly differentiate both sides of the equation with respect to time (T)
Finally, you must substitute all known variables into the resulting equation and solve for the desired rate of change
What are the key words you look for and what do they mean? Specific key words are used to recognize a related rate problem. YOUR KEY TERM or PHRASE IS: A related rate problem is one that involves the key words “something per something.” I.E. Gallons per minute Cubic feet per second
Your key word is PER Give an example of a related rate problem and solve it. Example: A Pebble is dropped into a calm pond causing ripples in the form of concentric circles. The radius of the outer circle is increasing at a rate of 1 foot PER second when the radius is 4 ft at what rate is the total area (A) of the disturbed water changing? #1 First, identify all given quantities and quantities to be determined. Also make a sketch and be sure to label the quantities.
dr/dt= 1ft/sec R=4 ft dA/dt= ?
#2 Second, you must choose and write an equation involving the variables whose rates of change either are given or are to be determined A= pi r ^2
#3 Third, you must use the chain rule implicitly differentiate both sides of the equation with respect to time (T)dA/dt= pi{2r dr/dt}
#4 Finally, you must substitute all known variables into the resulting equation and solve for the desired rate of change
dA/dt= 2pi (4)(1) Therefore the total area of the disturbed water is changing at a rate of 8 pi ft/sec Example: Air is being pumped into a spherical balloon at the radius of 4.5 cubic feet per minute. Find the rate if change of the radius when the radius is 2 ft #1 First, identify all given quantities and quantities to be determined. Also make a sketch and be sure to label the quantities.
Dv/dt= 4.5 ft^3/min R = 2 ft Dr/dt= ?
#2 Second, you must choose and write an equation involving the variables whose rates of change either are given or are to be determined V= 4/3 pi r^3 #3 Third, you must use the chain rule implicitly differentiate both sides of the equation with respect to time (T) Dv/dt= 4/3 pi [ 3r dr/dt] ^ = 4pi r^2 dr/dt #4 Finally, you must substitute all known variables into the resulting equation and solve for the desired rate of change
4.5 = 4 pi (2)^2dr/dt 4.5= 16 pi dr/dt 4.5/16= dr/dt Dr/dt= 9/32 pi ft/min Therefore the rate of change of the radius is 9/32 pi ft/min
The steps to solving a related rate problem are as follows: 1.) Identify what quantities and variables are given to you in the word problem. Label each one so you know what you have. Then make sure you know what you have to solve for. If a diagram/drawing can be made, take advantage of that, and label all parts of the drawing so that you can visualize what you have. 2.)Find an equation or formula you will have to use to solve the problem. This equation typically involves a shape described in a problem (not always a shape though..) and variables. And this equation will eventually be derived in the process of solving the problem. 3.) Derive all necessary parts of the problem using the chain rule, implicit differentiation, etc. (the equation/formula you found) 4.) After taking the derivative of said equation, you simply plug in all the values you know of to find the appropriate variable. *Keep in mind that whenever you get an answer, it MUST be followed by the appropriate UNITS (that is..if units are given in the problem)
KEY WORDSSSSSSSSSSSS :) *"Find the rate.." or "..rate of change.." = take DERIVATIVE!! *"with respect to.." = with respect to any variable besides x...you write the derivative as d*/dt..where * is the variable. *"radius" = "r" in whatever formula you're using..like for instance, A=pir^2 *"what rate is the total AREA...?" = since they're asking for a RATE, it will be expressed in the form d*/dt where * is A...So you'd be looking for dA/dt *"___cubic feet per minute" = (where ___ would be some number) "cubic feet"..or any measurement CUBED indicates VOLUME! *"rate of change of RADIUS" = again, since it's asking for RATE, they want the derivative expressed in the form dr/dt *height = depth = amplitude (^^ALL THE SAME THING!) *anything "rising" = indicates something to do with the height. So if they ask how HIGH something is rising or the RATE at which something is rising (both mean the same thing)..that means they're asking for the answer in the form dh/dt (h = height) *When dealing with triangles: dx/dt typically refers to the base of the triangle or x-axis...while dy/dt refers to the height of the triangle or y-axis...and "s" refers to the hypotenuse.
EXAMPLE: Volume...The radius r of a sphere is increasing at a rate of 3 inches per minute. Find the rates of change of the volume when r=9 inches and r=36 inches. *Alright, first let's see what the problem gave us to work with... First it says that the radius is increasing at a RATE of 3 in/min. Since this is the rate of "r" it can be expressed as dr/dt So dr/dt = 3 in/min Next they say to find the rates of change...(means you WILL have to DERIVE something)...of the VOLUME-->> meaning that you will have to use the volume formula for a sphere which is: V = 4/3pi(r)^3 ..(so you're looking for dV/dt) And since the problem tells you the radii (<I think that's the plural form of radius)..all that means is they want you to plug that in for "r" after you derive the formula *So first, you can derive the equation and you get: dV/dt = 4pi[r^2dr/dt] *Now you plug in the first radius they gave you (r=9) and dr/dt (which is 3) and you should get 972pi in^3/min (*remember, it's cubed because volume is always cubed) *Then you plug in the second radius (36) along with 3 (dr/dt) and you should get 15552 in^3/min
There are steps that you can take to decipher related rate word problems and make them easier to understand. First, you need to identify all of the information the word problem gives, such as variables and what you’re actually trying to find. Once you do that, then you need to find a formula that connects all of your variables and your shape together (such as the area of a triangle or the volume of a cube). After this, solve for your desired rate with respect to t (such as dA/dt for the rate of area changing or dV/dt for the rate of volume changing). And lastly, plug in all of your variables to get your answer. *Don’t forget to include your units (such as cm/min or ft^3/sec).
Some key words you need to know are: “rate of change” = derivative “speed” = derivative “with respect to” = what’s the bottom variable of the derivative (ex. rate of changing area A with respect to time t : dA/dt) “area” or “volume” = formula “cube” or “sphere” or “square” or “triangle” or “circle” = shape “radius” or “diameter” or “height” or “depth” = possibly what you need to take the derivate of or important components of the formula “increasing” or “decreasing” = whether the derivative is positive or negative and any form of units (especially squared or cubic units) = squared means area; cubic means volume
Ex. The radius r of a circle is increasing at a rate of 4 centimeters per minute. Find the rate of change of the area when r = 8 cm.
First of all, we know the following info: r = 8 & dr/dt = 4 cm/min We also know we are looking for: dA/dt = ? Second, because we are looking for the rate of change of the AREA, we know where using the area of a circle formula: A = π(r^2) Now, solve for dA/dt by taking the derivative of the area formula: dA/dt = 2π r dr/dt **Remember to put dr/dt after you take the derivative of r because it is an implicit derivative. Finally, plug in your variables: dA/dt = 2π (8)(4) = 64π cm^2/min ***cm is squared because area is always squared, just like volume is always cubed.
What are the steps to a related rate problem? -There are four main steps to solving a related-rate problem, and they can be referred to as the guidlines for solving a relate-rate problem. These are the four steps: Identify all given quanities and quantites to be determined. Make a sketch and label the quantities. Write an equation involving the variables whose rates of change either are given or are to be determined. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time (t). After completeing step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change. These steps are basically saying that first you have to figure out what each thing is in the problem, next your going to find the formula your dealing with, then take the derivative of the formula, plug in what you know to the problem, then you should get your answers. **Make sure you know the formulas before trying to work the problem. **Also the key words will help you.
Key Words:
The rate of something= dr/dt, dx/dt, dy/dt dt= with respect to time cubic feet= volume rate of change= derivative speed= derivative increasing or decreasing= if the derivative is changing **These are just a few
Example:
Air is being pumped into a balloon at the rate of 3 cubic feet per minute. Find the rate of change of the radius. When the radius is 2ft.
There are four steps to solving related rate problems.
1. Identify all given quantities and quantities to be determined. Make a sketch and label the quantities.
2. Write an equation involving the variables whose rates of change either are given or are to be determined.
3. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t.
4. After completing Step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.
What are the key words you look for and what do they mean?
Rate is a derivative with respect to a different variable.
Rate of something that is changing = dr/dt (rate radius is changing), dx/dt (rate x is changing), dy/dt (rate y is changing), dD/dt (rate distance is changing).
When dt is used, it means "with respect to time."
Give an example of a related rate problem and solve it.
A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer circle is increasing at a rate of 1 foot per second when the radius is 4 feet. At what rate is the total area A of the disturbed water changing?
What are the steps to a related rate problem?
ReplyDeleteThere are four major steps that can be used as a general guideline to solving a related rate problem however, it needs to be kept in mind that although there are set steps for working related rate problems, each problem is unique in its requirements for solving the problem completely.
The steps to solving these problems are as follows:
First, identify all given quantities and quantities to be determined. Also make a sketch and be sure to label the quantities.
Second, you must choose and write an equation involving the variables whose rates of change either are given or are to be determined
Third, you must use the chain rule implicitly differentiate both sides of the equation with respect to time (T)
Finally, you must substitute all known variables into the resulting equation and solve for the desired rate of change
What are the key words you look for and what do they mean?
Specific key words are used to recognize a related rate problem.
YOUR KEY TERM or PHRASE IS:
A related rate problem is one that involves the key words “something per something.”
I.E. Gallons per minute
Cubic feet per second
Your key word is PER
Give an example of a related rate problem and solve it.
Example:
A Pebble is dropped into a calm pond causing ripples in the form of concentric circles. The radius of the outer circle is increasing at a rate of 1 foot PER second when the radius is 4 ft at what rate is the total area (A) of the disturbed water changing?
#1 First, identify all given quantities and quantities to be determined. Also make a sketch and be sure to label the quantities.
dr/dt= 1ft/sec
R=4 ft
dA/dt= ?
#2 Second, you must choose and write an equation involving the variables whose rates of change either are given or are to be determined
A= pi r ^2
#3 Third, you must use the chain rule implicitly differentiate both sides of the equation with respect to time (T)dA/dt= pi{2r dr/dt}
#4 Finally, you must substitute all known variables into the resulting equation and solve for the desired rate of change
dA/dt= 2pi (4)(1)
Therefore the total area of the disturbed water is changing at a rate of 8 pi ft/sec
Example:
Air is being pumped into a spherical balloon at the radius of 4.5 cubic feet per minute. Find the rate if change of the radius when the radius is 2 ft
#1 First, identify all given quantities and quantities to be determined. Also make a sketch and be sure to label the quantities.
Dv/dt= 4.5 ft^3/min
R = 2 ft
Dr/dt= ?
#2 Second, you must choose and write an equation involving the variables whose rates of change either are given or are to be determined
V= 4/3 pi r^3
#3 Third, you must use the chain rule implicitly differentiate both sides of the equation with respect to time (T)
Dv/dt= 4/3 pi [ 3r dr/dt]
^ = 4pi r^2 dr/dt
#4 Finally, you must substitute all known variables into the resulting equation and solve for the desired rate of change
4.5 = 4 pi (2)^2dr/dt
4.5= 16 pi dr/dt
4.5/16= dr/dt
Dr/dt= 9/32 pi ft/min
Therefore the rate of change of the radius is 9/32 pi ft/min
The steps to solving a related rate problem are as follows:
ReplyDelete1.) Identify what quantities and variables are given to you in the word problem. Label each one so you know what you have. Then make sure you know what you have to solve for. If a diagram/drawing can be made, take advantage of that, and label all parts of the drawing so that you can visualize what you have.
2.)Find an equation or formula you will have to use to solve the problem. This equation typically involves a shape described in a problem (not always a shape though..) and variables. And this equation will eventually be derived in the process of solving the problem.
3.) Derive all necessary parts of the problem using the chain rule, implicit differentiation, etc. (the equation/formula you found)
4.) After taking the derivative of said equation, you simply plug in all the values you know of to find the appropriate variable.
*Keep in mind that whenever you get an answer, it MUST be followed by the appropriate UNITS (that is..if units are given in the problem)
KEY WORDSSSSSSSSSSSS :)
*"Find the rate.." or "..rate of change.." = take DERIVATIVE!!
*"with respect to.." = with respect to any variable besides x...you write the derivative as d*/dt..where * is the variable.
*"radius" = "r" in whatever formula you're using..like for instance, A=pir^2
*"what rate is the total AREA...?" = since they're asking for a RATE, it will be expressed in the form d*/dt where * is A...So you'd be looking for dA/dt
*"___cubic feet per minute" = (where ___ would be some number) "cubic feet"..or any measurement CUBED indicates VOLUME!
*"rate of change of RADIUS" = again, since it's asking for RATE, they want the derivative expressed in the form dr/dt
*height = depth = amplitude
(^^ALL THE SAME THING!)
*anything "rising" = indicates something to do with the height. So if they ask how HIGH something is rising or the RATE at which something is rising (both mean the same thing)..that means they're asking for the answer in the form dh/dt (h = height)
*When dealing with triangles:
dx/dt typically refers to the base of the triangle or x-axis...while dy/dt refers to the height of the triangle or y-axis...and "s" refers to the hypotenuse.
EXAMPLE:
Volume...The radius r of a sphere is increasing at a rate of 3 inches per minute. Find the rates of change of the volume when r=9 inches and r=36 inches.
*Alright, first let's see what the problem gave us to work with...
First it says that the radius is increasing at a RATE of 3 in/min. Since this is the rate of "r" it can be expressed as dr/dt
So dr/dt = 3 in/min
Next they say to find the rates of change...(means you WILL have to DERIVE something)...of the VOLUME-->> meaning that you will have to use the volume formula for a sphere which is:
V = 4/3pi(r)^3 ..(so you're looking for dV/dt)
And since the problem tells you the radii (<I think that's the plural form of radius)..all that means is they want you to plug that in for "r" after you derive the formula
*So first, you can derive the equation and you get: dV/dt = 4pi[r^2dr/dt]
*Now you plug in the first radius they gave you (r=9) and dr/dt (which is 3) and you should get 972pi in^3/min (*remember, it's cubed because volume is always cubed)
*Then you plug in the second radius (36) along with 3 (dr/dt) and you should get 15552 in^3/min
There are steps that you can take to decipher related rate word problems and make them easier to understand. First, you need to identify all of the information the word problem gives, such as variables and what you’re actually trying to find. Once you do that, then you need to find a formula that connects all of your variables and your shape together (such as the area of a triangle or the volume of a cube). After this, solve for your desired rate with respect to t (such as dA/dt for the rate of area changing or dV/dt for the rate of volume changing). And lastly, plug in all of your variables to get your answer.
ReplyDelete*Don’t forget to include your units (such as cm/min or ft^3/sec).
Some key words you need to know are:
“rate of change” = derivative
“speed” = derivative
“with respect to” = what’s the bottom variable of the derivative (ex. rate of changing area A with respect to time t : dA/dt)
“area” or “volume” = formula
“cube” or “sphere” or “square” or “triangle” or “circle” = shape
“radius” or “diameter” or “height” or “depth” = possibly what you need to take the derivate of or important components of the formula
“increasing” or “decreasing” = whether the derivative is positive or negative
and any form of units (especially squared or cubic units) = squared means area; cubic means volume
Ex. The radius r of a circle is increasing at a rate of 4 centimeters per minute. Find the rate of change of the area when r = 8 cm.
First of all, we know the following info: r = 8 & dr/dt = 4 cm/min
We also know we are looking for: dA/dt = ?
Second, because we are looking for the rate of change of the AREA, we know where using the area of a circle formula: A = π(r^2)
Now, solve for dA/dt by taking the derivative of the area formula: dA/dt = 2π r dr/dt
**Remember to put dr/dt after you take the derivative of r because it is an implicit derivative.
Finally, plug in your variables: dA/dt = 2π (8)(4) = 64π cm^2/min
***cm is squared because area is always squared, just like volume is always cubed.
What are the steps to a related rate problem?
ReplyDelete-There are four main steps to solving a related-rate problem, and they can be referred to as the guidlines for solving a relate-rate problem. These are the four steps:
Identify all given quanities and quantites to be determined. Make a sketch and label the quantities.
Write an equation involving the variables whose rates of change either are given or are to be determined.
Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time (t).
After completeing step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.
These steps are basically saying that first you have to figure out what each thing is in the problem, next your going to find the formula your dealing with, then take the derivative of the formula, plug in what you know to the problem, then you should get your answers. **Make sure you know the formulas before trying to work the problem. **Also the key words will help you.
Key Words:
The rate of something= dr/dt, dx/dt, dy/dt
dt= with respect to time
cubic feet= volume
rate of change= derivative
speed= derivative
increasing or decreasing= if the derivative is changing
**These are just a few
Example:
Air is being pumped into a balloon at the rate of 3 cubic feet per minute. Find the rate of change of the radius. When the radius is 2ft.
Step 1: dV/dt= 3 ft^3/min
dR/dt= ?
r=2
Step 2: V=4/3pir^3
dV/dt=4/3pi(3r^2dr/dt)
Step 3: 4pir^2(dr/dt)
Step 4: 3=4pi(2)^2dr/dt
3=16pidr/dt
ANSWER: 3/16ft/min=dr/dt
I don’t really understand the triangle problems like this. When drawing the triangle I’m not sure where which numbers go.
What are the steps to a related rate problem?
ReplyDeleteThere are four steps to solving related rate problems.
1. Identify all given quantities and quantities to be determined. Make a sketch and label the quantities.
2. Write an equation involving the variables whose rates of change either are given or are to be determined.
3. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t.
4. After completing Step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.
What are the key words you look for and what do they mean?
Rate is a derivative with respect to a different variable.
Rate of something that is changing = dr/dt (rate radius is changing), dx/dt (rate x is changing), dy/dt (rate y is changing), dD/dt (rate distance is changing).
When dt is used, it means "with respect to time."
Give an example of a related rate problem and solve it.
A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer circle is increasing at a rate of 1 foot per second when the radius is 4 feet. At what rate is the total area A of the disturbed water changing?
1. dr/dt = 1 ft/s, r=4 ft., dA/dt=?
2. A=pi(r^2)
3. dA/dt= pi(2r(dr/dt)) = 2pi(r)(dr/dt)
4. dA/dt= 2pi(4)(1) = 8pi ft/s <------FINAL ANSWER
1st. Find all of your given variables, and if the information is given draw the figure that is in the equation
ReplyDelete2nd. Find the right equation for the variables you are given, and an equation that will help you to identify your unknown.
3rd. Differentiate both sides of the equation using whatever methods neccessary.
4th. Plug in the values you know into the equation and solve for your unknown.
Ex. Suppose x and y are both differentiable functions of t and are related by the equation y=x^2+3. Find dy/t when x=2 given that dx/dt=4 when x=2.
y=x^2+3 Deriviate the equation
1dy/dt= 2x dx/dt+0
dy/dt= 2(2)(4)
dy/dt= 16
key words
rate of change = derivative
speed = derivative
with respect to = what’s the bottom variable of
the derivative
area = formula
volume = formula
cube = shape
sphere = shape
square = shape
triangle = shape
circle = shape
radius = take the derivative
diameter = take the derivative
height = take the derivative
depth = take the derivative
increasing = positive derivative
decreasing = negative derivative
units = squared means area and cubic means volume