# 4.1 Related rates  (Page 2/7)

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Note that when solving a related-rates problem, it is crucial not to substitute known values too soon. For example, if the value for a changing quantity is substituted into an equation before both sides of the equation are differentiated, then that quantity will behave as a constant and its derivative will not appear in the new equation found in step 4. We examine this potential error in the following example.

## Examples of the process

Let’s now implement the strategy just described to solve several related-rates problems. The first example involves a plane flying overhead. The relationship we are studying is between the speed of the plane and the rate at which the distance between the plane and a person on the ground is changing.

## An airplane flying at a constant elevation

An airplane is flying overhead at a constant elevation of $4000\phantom{\rule{0.2em}{0ex}}\text{ft}.$ A man is viewing the plane from a position $3000\phantom{\rule{0.2em}{0ex}}\text{ft}$ from the base of a radio tower. The airplane is flying horizontally away from the man. If the plane is flying at the rate of $600\phantom{\rule{0.2em}{0ex}}\text{ft/sec},$ at what rate is the distance between the man and the plane increasing when the plane passes over the radio tower?

Step 1. Draw a picture, introducing variables to represent the different quantities involved.

As shown, $x$ denotes the distance between the man and the position on the ground directly below the airplane. The variable $s$ denotes the distance between the man and the plane. Note that both $x$ and $s$ are functions of time. We do not introduce a variable for the height of the plane because it remains at a constant elevation of $4000\phantom{\rule{0.2em}{0ex}}\text{ft}.$ Since an object’s height above the ground is measured as the shortest distance between the object and the ground, the line segment of length 4000 ft is perpendicular to the line segment of length $x$ feet, creating a right triangle.

Step 2. Since $x$ denotes the horizontal distance between the man and the point on the ground below the plane, $dx\text{/}dt$ represents the speed of the plane. We are told the speed of the plane is 600 ft/sec. Therefore, $\frac{dx}{dt}=600$ ft/sec. Since we are asked to find the rate of change in the distance between the man and the plane when the plane is directly above the radio tower, we need to find $ds\text{/}dt$ when $x=3000\phantom{\rule{0.2em}{0ex}}\text{ft}.$

Step 3. From the figure, we can use the Pythagorean theorem to write an equation relating $x$ and $s\text{:}$

${\left[x\left(t\right)\right]}^{2}+{4000}^{2}={\left[s\left(t\right)\right]}^{2}.$

Step 4. Differentiating this equation with respect to time and using the fact that the derivative of a constant is zero, we arrive at the equation

$x\frac{dx}{dt}=s\frac{ds}{dt}.$

Step 5. Find the rate at which the distance between the man and the plane is increasing when the plane is directly over the radio tower. That is, find $\frac{ds}{dt}$ when $x=3000\phantom{\rule{0.2em}{0ex}}\text{ft}.$ Since the speed of the plane is $600\phantom{\rule{0.2em}{0ex}}\text{ft/sec},$ we know that $\frac{dx}{dt}=600\phantom{\rule{0.2em}{0ex}}\text{ft/sec}.$ We are not given an explicit value for $s;$ however, since we are trying to find $\frac{ds}{dt}$ when $x=3000\phantom{\rule{0.2em}{0ex}}\text{ft},$ we can use the Pythagorean theorem to determine the distance $s$ when $x=3000$ and the height is $4000\phantom{\rule{0.2em}{0ex}}\text{ft}.$ Solving the equation

${3000}^{2}+{4000}^{2}={s}^{2}$

for $s,$ we have $s=5000\phantom{\rule{0.2em}{0ex}}\text{ft}$ at the time of interest. Using these values, we conclude that $ds\text{/}dt$ is a solution of the equation

$\left(3000\right)\left(600\right)=\left(5000\right)·\frac{ds}{dt}.$

Therefore,

$\frac{ds}{dt}=\frac{3000·600}{5000}=360\phantom{\rule{0.2em}{0ex}}\text{ft/sec}.$

Note : When solving related-rates problems, it is important not to substitute values for the variables too soon. For example, in step 3, we related the variable quantities $x\left(t\right)$ and $s\left(t\right)$ by the equation

${\left[x\left(t\right)\right]}^{2}+{4000}^{2}={\left[s\left(t\right)\right]}^{2}.$

Since the plane remains at a constant height, it is not necessary to introduce a variable for the height, and we are allowed to use the constant 4000 to denote that quantity. However, the other two quantities are changing. If we mistakenly substituted $x\left(t\right)=3000$ into the equation before differentiating, our equation would have been

${3000}^{2}+{4000}^{2}={\left[s\left(t\right)\right]}^{2}.$

After differentiating, our equation would become

$0=s\left(t\right)\frac{ds}{dt}.$

As a result, we would incorrectly conclude that $\frac{ds}{dt}=0.$

what is function?
A set of points in which every x value (domain) corresponds to exactly one y value (range)
Tim
what is lim (x,y)~(0,0) (x/y)
limited of x,y at 0,0 is nt defined
Alswell
But using L'Hopitals rule is x=1 is defined
Alswell
Could U explain better boss?
emmanuel
value of (x/y) as (x,y) tends to (0,0) also whats the value of (x+y)/(x^2+y^2) as (x,y) tends to (0,0)
NIKI
can we apply l hospitals rule for function of two variables
NIKI
why n does not equal -1
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
proof the formula integration of udv=uv-integration of vdu.?
Find derivative (2x^3+6xy-4y^2)^2
no x=2 is not a function, as there is nothing that's changing.
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
y=800
800
Bg
how do u factor the numerator?
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
2x^3+6xy-4y^2)^2 solve this
femi
moe
volume between cone z=√(x^2+y^2) and plane z=2
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
what is a function? f(x)
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
is x=2 a function?
The
What is limit
it's the value a function will take while approaching a particular value
Dan
don ger it
Jeremy
what is a limit?
Dlamini
it is the value the function approaches as the input approaches that value.
Andrew
Thanx
Dlamini
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
Antonio
is x=2 a function?
The