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
Homework Statement
Stewart Calculus 6E, 3.8 #14
At noon, ship A is 150km west of ship B. Ship A is sailing east at 35km/h, and ship B is sailing north at 25km/h. How fast is the distance between the ships changing at 4pm?
Homework Equations
Distance Formula I think.
The Attempt at a...
Ok, I've been trying to figure this problem out of an old textbook for a couple days but seem to get nowhere.
"A swimming pool with a rectangular surface 18 meters long and 12 meters wide is being filled at the rate of .80m^3/min. At one end it is 1.0 m deep and at another it is 2.5 m deep...
Homework Statement
A ladder of 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away at 2 ft per second.
(a) How fast is the top of the ladder moving down the wall when the base is seven feet from the wall?
(b) Consider the triangle formed by the...
Homework Statement
Dear All,
I am having problems understanding how to deal with related rates. The problem is the following:
A solid 400gm metal cube of size length 10cm expands uniformly when heated. If the length of its side expand at 0.5cm(hr), find the rate at which, after 5...
Homework Statement
A 6-ft man walks away from a 15-ft lamppost at a speed of 3 ft/s. Find the rate at which his shadow is increasing in length.
Homework Equations
Pythagorean Theorem, Theorem of Similar Triangles (ratios of corresponding sides are equal)
The Attempt at a Solution...
Homework Statement
The number of yeast cells in a laboratory culture increases rapidly at first but levels off eventually. The population is modeled by the function below
n=f(t)=\frac{a}{1+be^{-t}}
where t is measured in hours. at time t=0 the population is 20 cells and is increasing...
Homework Statement
Air is being pumped into a spherical balloon so that its volume increases at a rate of 70 cm3/s. How fast is the surface area of the balloon increasing when its radius is 5 cm3?Homework Equations
A ball with radius r has the following volume (V) and surface area (S):
V =...
Homework Statement
If there is a container with a radius in feet given by r(z)=10/sqrt(z) and it is being filled at a constant rate of 22pi ft^3/min. find a function that gives the height of the water as a function of time.The Attempt at a Solution
since the radius is a function of the...
Homework Statement
A piston is seated at the top of a cylindrical chamber with radius 5 cm when it starts moving into the chamber at a constant speed of 3 cm/s. What is the rate of change of the volume of the cylinder when the piston is 2cm from the base of the chamber?
Homework...
Homework Statement
I am reading about related rates in my calc book but it doesn't really explain it very well. Are there any arcitcals some one can send me that goes indepth about this concept on related rates?
Thank you.
Homework Statement
Fluid flows out of a cone at 3 cu ft/min. If the height of the cone is three times the radius how fast is the height of the fluid decreasing when the fluid is 6 inches deep in the middle?
The Attempt at a Solution
1. V= ( (pi)(r^2)(h) ) / 3
2. r = h/3 thus... V =...
Okay so I have this related rates problem for my AP Calculus Class
A trough (Extruded Trapezoid) with a height of 2, Base 1 (bottom) of 2, Base 2 (top) of 6 and a extrusion of 10.
(I provided a picture for better understanding)
--------------------------------------…
A = Area of the top...
Gravel is being dumped from a conveyor belt at a rate of 30 ft3/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?
Known:
dV/dt=30 ft^3/min...
Related Rates Promlem, Please Help!
Homework Statement
A bird of prey is perched at the top of a tree that is 40 m high. The bird watches as a delectable Kingston squirrel runs away from the base of the tree at a rate of 2 m/s. What is the rate of change of the distance between the bird...
Homework Statement
At a certain moment, a plane passes directly above a radar station at an altitude of 3 miles at a speed of 300 mph. Suppose that the line through the radar station and the plane makes an angle θ with the horizontal. How fast is θ changing 50 min after the plane passes over...
Homework Statement
The volume of a spherical balloon is increasing at a rate of 4m^3/min. How fast is the surface area increasing when the radius is three meters?
Homework Equations
V=4/3piR^3
A=4piR^2
The Attempt at a Solution
V=s.a.*R/3
dv/dt=d(s.a.R/3)/dt...
Homework Statement
Two sides of a triangle of fixed length measure 12m and 15m. The angle between these two sides increases at a rate of 2 degrees per minute. When the angle between these two sides is 60 degrees, at what rate is the third side increasing?
Homework Equations
That's the...
Homework Statement
A ladder, of length 13m, is leaning against a wall. The base of the ladder begins to slide, frictionless, away from the wall; by the time the base is horizontally 12m away from the wall it is moving at a rate of 2ms-1
a)How fast is the top of the ladder sliding down the...
URGENT - related rates pool problem
Homework Statement
A swimming pool with a rectangular surface 18 m long and 12m wide is being filled at the rate of 0.8 cubic metres/min. At one end the pool is 1m deep and at the other end it is 2.5m deep, with a constant slope between the ends. How fast...
Homework Statement
[PLAIN]http://img638.imageshack.us/img638/2218/questionscreenshot.jpg
The attempt at a solution
Step 1. Draw a triangle. Solve for third side (Pythagorean theorem), which I called z.
z^2 = x^2 + y^2
Coordinates are given (4, 6). We know x = 4, y = 6...
Homework Statement
2 Cylindrical pools are filled simultaneously at the same rate, 1m3/min. The smaller pool has radius 5m and the water level rises at a rate of 0.5m/min. The larger pool has a radius 8m. How fast is the water level rising in the larger pool?
Homework Equations
V =...
Homework Statement
I was really struggling with a Calculus problem but I think I have the right answer, this is more just checking with someone than anything.
When given a Ferris Wheel has a radius of 10 m, and that it takes 2 minutes for one rotation, find how fast it is moving from the...
When trying to solve a related rates problem, sometimes you need to relate variables. Can anyone help me with the conceptual reasoning or at least what to look for in these problems?
Homework Statement
A pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. If the pool is shaped as shown (I attached a picture), and is being filled at a rate of 8 ft^3/min, how fast is the water level rising when the depth at the deepest point is...
Homework Statement
"Consider the tank and water supply system as shown in the figure below. The diameter
of the supply pipe, D1 = 20 mm, and the average velocity leaving the supply pipe is
V1 = 0.595 m/s. A shut-off valve is located at z = 0.1 m, and the exit pipe, D2 = 10 mm.
The tank...
Please answer this, it's so hard.
Problem:
At 12 noon, ship A travels due west at 100 knotsshiop B. After 4hours, ship B leaves the same port on the direction N30'W at 80 knots. How fast is the distance between two ships changing at 8PM in the same day?
Reply please.I can't solve it...
Water is emptied from a spherical tank of radius 24 ft. The water in the tank is 8 ft deep and its level is decreasing at the rate of 2 ft/ min. At this time, what is the rate of change of the radius at the top of the water?
Homework Statement
A kite 100 ft above the ground moves horizontally at a speed
of 8 ft!s. At what rate is the angle between the string and the
horizontal decreasing when 200 ft of string has been let out?
Homework Equations
tan \o = x/y
The Attempt at a Solution
After...
One night, a toddler is dropping eggs from a windowof a building which is 20 feet above ground. A neighbor, hearing the commotion, looks through her window which is 15 feet from the toddler's window, on the same wall of the building, and at the same height. She shines a flashlight from her...
Homework Statement
A swimming pool is 20 feet wide, 40 feet long, 3 feet deep at the shallow end and 9 feet deep at its deepest point. If the pool is being filled at a rate of 0.8 feet^3/min, how fast is the water level rising when the depth at the deepest point is 5 feet?
Homework Equations...
Homework Statement
An inverted conical container has a diameter of 42 in and a depth of 15 in. If water is flowing out of the vertex at a rate of 35 \pi in^3 / sec, how fast is the depth of the water dropping when the height is 5 in?
Homework Equations
v= (1/3) \pi r^2 * h
The...
Homework Statement
Two boats leave a dock at the same tie. One goes south at 30 km/h and the other goes southwest at 40 km/h. What is the rate of change of the distance between the 2 boats, 30 minutes after they leave?
I know how to do these questions when it's a right triangle...but since...
Homework Statement
Water is leaking out an inverted conical tank at the rate of 50L/min. The tank has a diameter of 10m at the top and is 6m deep. if the water is rising at the rate of 4cm/min when the greatest depth is 3m, find the rate at which the water is being poured into the tank...
Hi, I've tried this too many ways and i can't seem to figure it out. the question is:
As sand leaks out of a hole in a container, it forms a conical pile whose altitude is always the same as its radius. If the height of the pile is increasing at a rate of 6in/min, find the rate at which the...
Hi, I've tried this too many ways and i can't seem to figure it out. the question is:
As sand leaks out of a hole in a container, it forms a conical pile whose altitude is always the same as its radius. If the height of the pile is increasing at a rate of 6in/min, find the rate at which the...
Homework Statement
A spherical tank with an interior diameter of 2m is being filled with water at a rate of 10 liters per second. Determine what rate the height of the water is increasing at when the tank is half full.
Homework Equations
V=\frac{2\pi r^3}{3}-\pi r^2d+\frac{\pi d^3}{3}...
Homework Statement
At ten am, a plane traveling east at 100km/hr is 500 km west of a jet traveling south at 200km/hr. At what rate is the distance between them changing at noon?
Homework Equations
The Attempt at a Solution
2aa' + 2bb' = 2cc'
2(300)(-100) + 2(400)(200) = (1000)c'...
Homework Statement
Water si emptying out of a tank at a rate of 4 cubic metres/min. The tank is shaped like an inverted circular cone with a diameter of 8 m at the top. The sides of the cone make a 30 degree with the vertical. Determine the exact value of the rate of change of the water level...
Homework Statement
A man 6m tall walks away from a light 10m above the ground. If his shadow lengthens at a rate of 2m/s, how fast is he walking?
Homework Equations
The Attempt at a Solution
I have nothing.
Homework Statement
A conical reservoir is 50m deep and 200m across the top. Water is being pumped in at a rate of 7000m^3/min, and at the same time, water is being drained out at a rate of 9000 m^3/min. What is the change in diameter when the height of the water in the reservoir is 20m...
Homework Statement
A trough is 9 ft long and its ends have the shape of isosceles triangles that are 5 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 14 ft3/min, how fast is the water level rising when the water is 8 inches deep...
Homework Statement A swimming pool is 5 m wide, 10 m long, 1 m deep at the shallow end, and 3 m deep at its deepest point. A cross-section is shown in the figure. If the poole is being filled at a rate of 0.1 m^3 per minute, how fast is the water level rising when the depth at the deepest...
Homework Statement
A lighthouse is located off shore one mile from the nearest point P, on a straight coastline. The light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 2 miles from point P?
Homework Equations
The Attempt at a...
Hi I was just curious as to why calculus does not seem to work in the situation of area calculation. I came across this doing some HW and noticed that the booked worked out a seemingly similar question in two separate ways
First was predicitng the chance in the length of the hypotenuse...
I have a question about a related rates, i already solved it, but i would like someone to confirm it to me .
* The relation between distance s and velocity v is given by v=\frac{150s}{3+s}. Find the acceleration in terms of s.
As you can see my final answer is a in terms of v and s, not s only...
Homework Statement
A magic stalk starts growing at a spot 10 metres from a 4 metre lamppost. The stalk grows 1 metre per hour. Find the rate at which the length 's' of the stalk's shadow is increasing at the instant when the height h of the stalk is 3 metres.
The Attempt at a Solution...
Homework Statement
Water is being drained from a container which has the shape of an inverted right circular cone. The container has a radius of 5.00 inches at the top and a height of 7.00 inches. At the instant when the water in the container is 4.00 inches deep, the surface level is...
Homework Statement
A swimming pool is 40 feet long, 20 feet wide, 4 feet deep at the shallow end, and 9 feet deep at the deep end. Water is being pumped into the pool at 10 cubic feet per minute.
a. When the water is 3 feet deep at the shallow end, at what rate is the water level rising? b...
Homework Statement
"An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other. One plane is 120 miles from the point and is moving at 400 miles per hour. The other plane is 160 miles from the point and has a speed of...