







Sipping juice from a Tetra Pak 
20200211 

Anjhelic pose la question : Karen is sipping juice from a 1 in. by 3 in. by 6 in tetra pack at the rate of 0.5in³/sec. How fast is the height of juice in the pack decreasing? Harley Weston lui répond. 





Related rates 
20180211 

angelo pose la question : hi admin please help me answer this question. thank you!
At a certain instant of time, the angle A of a triangle ABC is 60 degrees and increasing at the rate of 5degrees per second, the side AB is 10cm and increasing at the rate of 1cm per second, and side AC is 16cm and decreasing at the rate of 1/2 cm per second. Find the rate of change of side AB? Penny Nom lui répond. 





Heat equation 
20171123 

Max pose la question : What does du\dt=a(triangle)^2u mean. Can it be solved for t. Penny Nom lui répond. 





How many miles did he drive in one hour? 
20171121 

Ava pose la question : Graham drove 42 1/3 Miles in 1 1/3 hours. How many miles did he drive in one hour? Penny Nom lui répond. 





The distance between the origin and a moving point on a graph 
20171016 

Paulina pose la question : Find the rate of change of the distance between the origin and a moving point on the graph of y=x^2 +1 if dx/dt=2 centimeters per second Penny Nom lui répond. 





The average rate of change of cot(t) 
20170118 

Brianna pose la question : Hello!
It's been a while since I've taken a math course, and I'm stuck on a problem in my calculus course.
The question is this:
Find the average rate of change of the function over the given interval.
h(t)=cot(t) a) [5pi/4, 7pi/4] Penny Nom lui répond. 





Water leaking from a trough 
20161228 

Kathryn pose la question : A trough is 6 m long, and has uniform crosssection of an equilateral triangle with sides 1 m.
Water leaks from the bottom of the trough, at a constant rate of 0.1 m3/min.
Find the rate at which the water level is falling when the water is 0.2m deep. Penny Nom lui répond. 





Related rates 
20130217 

Ishaak pose la question : A hemispherical bowl is filled with water at a uniform rate. When the height of water is h cm the volume is π(rh^21/3 h^3 )cm^3, where r s the radius. Find the rate at which the water level is rising when it is half way to the top, given that r = 6 and the bowl fills in 1 minute. Penny Nom lui répond. 





How fast is the distance between the aircraft and the car increasing? 
20121024 

Steven pose la question : At a certain instant an aircraft flying due east at 240 miles per hour passes directly over a car traveling due southeast at 60 miles per hour on a straight, level road. If the aircraft is flying at an altitude of .5mile, how fast is the distance between the aircraft and the car increasing 36 seconds after the aircraft passes directly over the car? Penny Nom lui répond. 





The spread of a rumor 
20120409 

Roohi pose la question : The function f(t) = a/(1+3e^(bt)) has also been used to model the spread of a rumor. Suppose that a= 70 and b=3 0.2. Compute f(2), the percentage of the population that has heard the rumor after 2 hours. Compute f'(2) and describe what it represents. Compute lim t approaches infinity and describe what it represents. Penny Nom lui répond. 





The period T of a pendulum 
20120327 

Ashley pose la question : The period T of a pendulum is given in terms of its length, l, by T=2pi sqrt(l/g) where g is the acceleration due to gravity(a constant)
a. find dT/dl
b. what is the sign of dT/dl
c. what does the sign of dT/dl tell you about the period of the pendulums? Penny Nom lui répond. 





Water is flowing into a cup 
20111219 

Tim pose la question : A cup has a radius of 2" at the bottom and 6" on the top. It is 10" high. 4 Minutes ago, water started pouring at 10 cubic " per minute. How fast was the water level rising 4 minutes ago? How fast is the water level rising now? What will the rate be when the glass is full? Penny Nom lui répond. 





A cube of ice is melting 
20111205 

Emily pose la question : a cube of ice (i.e.) each side is of the same length) is melting at a rate such that the length of each side is decreasing at a rate of 5cm per hour. how fast is the volume of the cube decreasing (in cubic cm per hour) at the instant the length of each side is 25cm? Penny Nom lui répond. 





Water pouring into a conical tank 
20111121 

Patience pose la question : Hi my name is patience and I'm having a problem with this question.
Water pours into a conical tank of semi vertical angle 30 degrees at the rate of 4 cm^3/s, where h is the depth of the water at time t. At what rate is the water rising in the tank when h = 10 cm?
Thank you Penny Nom lui répond. 





A reservoir has the shape of an inverted cone 
20111003 

Roger pose la question : a reservoir has the shape of an inverted cone whose cross section is an equilateral triangle. if water is being pumped out of the reservoir at a rate of 2m^3/sec, at what rate is the depth of the water changing when the depth is 40 meters? Penny Nom lui répond. 





A hemispherical bowl with a lead ball inside 
20110927 

Jean pose la question : "(a) Water is being poured into a hemispherical bowl of radius 3 inch
at the rate of 1 inch^3/s. How fast is the water level rising when the
water is 1 inch deep ?
(b) In (a), suppose that the bowl contains a lead ball 2 inch in
diameter, and find how fast the water level is rising when the ball is
half submerged." Penny Nom lui répond. 





The height of a fluid in a horizontal tank 
20110724 

jason pose la question : Same set up as many others, cylindrical tank on its side, but I am interested in defining the change in volume and/or fluid level as a function of time at a constant volumetric outflow. I plan on hooking a pump to the tank so "gpms' will be constant. I have a couple different sized tanks and pumps so I want a general equation. Thanks for your help. Harley Weston lui répond. 





A conical container and a spherical balloon 
20110406 

Steven pose la question : Water is running out of a conical container 12 feet in diameter and 8 feet deep (vertex down) and filling a spherical balloon.
At the instant the depth of the water in the cone is 4 feet, the radius of the sphere is approximately 4 feet.
The rate of change of the depth of the water in the cone at the instant is approximately ______________ times the rate of change of the radius of the balloon. Penny Nom lui répond. 





Two ships 
20110405 

Gevork pose la question : Ship A is sailing due south at 16 mph. At the same time, a second ship B, 32 miles south of A, is sailing due east at 12 mph.
(a) at what rate are they approaching or separating at the end of one hour?
(b) At what rate are they approaching or separating at the end of two hours?
(c) When do they cease to approach each other and how far apart are they at that instant. Penny Nom lui répond. 





The rate of change of the area of a parallelogram 
20110405 

Gevork pose la question : Let a parallelogram have sides of 8 and 12 and let vertex angle A be decreasing at a rate of 2degrees per minute. Find the rate of change of the area of the parallelogram when angle A equals 30 degrees. Penny Nom lui répond. 





The rate of change of (8e^3x)+(27 e^3x) 
20101123 

Aleo pose la question : I am unable to solve this problem:
Find the rate of change of (8e^3x)+(27 e^3x), with respect to x when x= 0.5 Penny Nom lui répond. 





The distance between the origin and a moving point 
20100924 

Norma pose la question : I am having problems with this question
find the rate of change of the distance between the origin and a moving point on the graph of the function below if dx/dt=5 cm/sec
y=x^2+2 Penny Nom lui répond. 





The rate of change of y with respect to x 
20100429 

Tom pose la question : I just had a quick calc question about wording that wasn't ever
addressed in class. When the book says "the rate of change of y with
respect to x", should it be considered how fast y is changing in
comparison to x?
I ask because the textbook says that "y is changing 3 times faster than x,
so the rate of change of y with respect to x is 3." I'm use to rate being
like velocity, as in units of distance per units of time. All we're told
in class is that it's the slope of the tangent line, I was hoping you
could clarify for me what exactly is meant by the wording of a "rate of
change of something with respect to something else". More specifically, what
"rate" and "with respect to" mean within this context?
Thanks for your time Harley Weston lui répond. 





At what rate are the people moving apart? 
20091101 

saira pose la question : A man starts walking north at 4 ft/s from a point P. 5 minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 minute after the woman starts walking ? Harley Weston lui répond. 





The rate of change of the volume of a sphere 
20090325 

Kaylin pose la question : why the rate of change of volume of a sphere is not constant even though dr/dt is constant? Walter Whiteley lui répond. 





Related rates 
20090314 

Jeevitha pose la question : The side of an equilateral triangle decreases at the rate of 2 cm/s.
At what rate is the area decreasing when the area is 100cm^2? Stephen La Rocque lui répond. 





How fast is the visible surface of the earth decreasing? 
20090124 

Ray pose la question : A dive bomber loss altitude at a rate of 400 mph. How fast is the visible surface of the earth decreasing when the bomber is one mile high? Harley Weston lui répond. 





In the shadow of a flagpole 
20090122 

La pose la question : How fast is the length of the shadow of an 18 foot flagpole growing when the angle of elevation of the sun is 45 degrees and is decreasing at a rate of 10 degrees per hour? Harley Weston lui répond. 





Negative rate of change 
20090112 

hemanshu pose la question : when i have to find rate of change of decrease in any value my ans comes in negative why?????????? Penny Nom lui répond. 





Related rates 
20081126 

Lyudmyla pose la question : How fast is the volume of a cone increasing when the radius of its base is 2 cm and growing at a rate of 0.4 cm/s, and its height is 5 cm and growing at a rate of 0.1 cm/s? Harley Weston lui répond. 





How fast is the distance between the airplanes decreasing? 
20081110 

Crystal pose la question : At a certain instant, airplane A is flying a level course at 500 mph. At the same time, airplane B is straight above airplane A and flying at the rate of 700 mph. On a course that intercepts A's course at a point C that is 4 miles from B and 2 miles from A. At the instant in question, how fast is the distance between the airplanes decreasing? Harley Weston lui répond. 





Melting ice on a hemisphere 
20081020 

heather pose la question : The top of a silo is the shape of a hemishere of diameter 20 ft. if it is coated uniformly with a layer of ice, and if the thickness is decreasing at a rate of 1/4 in/hr, how fast is the volume of ice changing when the ice is 2 inches thick? Penny Nom lui répond. 





Related rates 
20081016 

Gisela pose la question : 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 6 in/min, find the rate at which the sand is leaking out when the
altitude is 10in. Penny Nom lui répond. 





The rate of change of the volume of a cone 
20081015 

Barbara pose la question : Suppose that both the radius r and height h of a circular cone change at a rate of 2 cm/s.
How fast is the volume of the cone increasing when r = 10 and h = 20? Harley Weston lui répond. 





The average rate of change of gasoline used 
20081006 

JHulie pose la question : What is the average rate of change of gasoline used, measured in miles per gallons
if you travel 212 miles, then you fill your gas tank up again and it takes 10.8 gallons.
If you designate your change in distance as 212 miles and your change in gallons as 10.8? Penny Nom lui répond. 





Trough Filling with Water 
20080821 

lanny pose la question : a triangular trough is 10 feet long, 6 feet across the top, and 3 feet deep. if water flows at the rate of 12 cubic inches per minute, find how fast the surface is rising when the water is 6 inches deep. Janice Cotcher lui répond. 





Slope and rate of change 
20080623 

Lee pose la question : What is the difference between a slope and a rate of change? Stephen La Rocque lui répond. 





The rate of change in the depth of the water 
20080612 

Liz pose la question : A rectangular pool 50ft long and 30ft. wide has a depth of 8 ft. for the first 20 ft. for its length and a depth of 3 ft. on the last 20ft. of its length and tapers linearly for the 10 ft in the middle of its length. the pool is being filled with water at the rate of 3ftcubed/ min
at what rate is the depth of the water in the pool increasing after 15 hours? Harley Weston lui répond. 





The average rate of change 
20080329 

Tom pose la question : For the function x/3x1 find the average rate of change between the interval x=1 and x=5? Harley Weston lui répond. 





The rate of change of the concentration of a solution 
20071030 

Nicholas pose la question : A barrel initially has two kg of salt dissolved in twenty liters of water. If water flows in the rate of 0.4 liters per minute and the wellmixed salt water solution flow out at the same rate, how much salt is present after 8 minutes?
I tried working backwards given the answer but I can't seen to get their answer of ~1.7kg. Any help would be great! Thanks Harley Weston lui répond. 





The rate of change of the area of a triangle 
20071022 

Ahlee pose la question : So my question is:
The included angle of the two sides of a constant equal length s of an isosceles triangle is ϑ.
(a) Show that the area of the triangle is given by A=1/2s^2 sinϑ
(b) If ϑ is increasing at the rate of 1/2 radian per minute, find the rate of change of the area when ϑ=pi/6 and ϑ=pi/3.
(c) Explain why the rate of change of the area of a triangle is not constant even though dϑ/dt is constant Penny Nom lui répond. 





The average rate of change of a function 
20071011 

vern pose la question : Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. f(X)=sinX for the inverval [0,pi/6]? Harley Weston lui répond. 





Rate of change of distance between the clock hands 
20070605 

Jonathan pose la question : A certain Clock has a minute hand with a length of 4 inches long and an hour hand with a length of 3 inches long. How fast is the distance between tips of these hands changing at 9:00? Stephen La Rocque lui répond. 





At what rate is the area of the triangle changing? 
20070224 

mac pose la question : two sticks 3.5 feet long are hinged together and are stood up to form an isosceles triangle with the floor. The sticks slide apart, and at the moment when the triangle is equilateral, the angle is increasing at the rate of 1/3 radian/sec. At what rate is the area of the triangle increasing or decreasing at that moment? Mac Penny Nom lui répond. 





A particle moving along a curve 
20061118 

Rachel pose la question : a particle is moving along the curve whose equation is: (xy^3)/(1+y^2)=8/5 assume the xcoordinate is increasing at the rate of 6 units per second when the particle is at the point (1,2). a. at what rate is the ycoordinate of the point changing at that instant? b. is the particle rising or falling at that instant? Penny Nom lui répond. 





An aircraft and a missile 
20061118 

Sarah pose la question : an aircraft is flying at a constant altitude with a constant speed of 600mph. an antiaircraft missile is fired on a straight line perpendicular to the flight path of the aircraft so that it will hit the aircraft at a point P. at that instant the aircraft is 2 miles from the impact point P the missile is 4 miles from P and flying at 1200 mph. at that instant, how rapidly is the distance between missile and aircraft decreasing? Stephen La Rocque lui répond. 





Wheat is poured on a conical pile 
20061117 

Rachel pose la question : wheat is poured through a chute at the rate of 10 cubic feet per minute and falls in a conical pile whose bottom radius is always half the altitude. how fast will the circumference of the base be increasing when the pile is 8 feet high? Penny Nom lui répond. 





The rate of change of the perimeter of a square 
20061107 

Karli pose la question : Find the rate of change of the perimeter of a square with respect to its area. Stephen La Rocque lui répond. 





A boat is being pulled towards a dock. 
20061106 

Cassie pose la question : A boat is being pulled towards a dock. If the rope is being pulled in at 3 feet per second, how fast is the distance between the dock and the boat decreasing when it is 30 feet from the dock? Penny Nom lui répond. 





How fast is the water level rising when the water is 1 meter deep? 
20061019 

Don pose la question : The cross section of a 5meter trough is an isosceles trapezoid with a 2meter lower base, a 3meter upper base and an altitude of 2 meters. Water is running into the trough at a rate of 1 cubic meter per minute. How fast is the water level rising when the water is 1 meter deep? Stephen La Rocque lui répond. 





The velocity of a pendulum, part II 
20060907 

Erin pose la question : We saw the question in your database about the velocity of a pendulum swinging.....It is the same exact question....but there is another question......it says....
"estimate the instantaneous rate of change of d with respect to t when t = 1.5. At this time, is the pendulum moving toward or away from the wall? Explain." Harley Weston lui répond. 





A rate of change problem 
20041015 

Frank pose la question : Find the rate of change of the distance between the origin and a moving point on the graph of y = x(squared) + 1 if dx/dt = 2 centimeters per second. Penny Nom lui répond. 





Rate of change problems 
20040801 

Jim pose la question : I just want to check a couple average rate of change problems because i just guessed on how to do them. Can you tell me how to do them?
the question says if f(x) = sqrt(x + 3), find f( x + rx). I got sqrt( x + rx + 3)
the other two are : f(x)= 3x1 (f(x)  f(1)) / (x1) ... I GOT 3 &
f(x)= x^3  x (f(x)  f(1)) / (x1) ... I GOT x^2 + x Penny Nom lui répond. 





A spotlight shines on a wall 
20020525 

Barb pose la question : A spotlight on the ground shines on a wall 12m away. If a man 2m tall walks from the spotlight toward the bldg at a speed of 1.6 m/s, how fast is his shadow on the bldg decreasing when he is 4m from the bldg? Penny Nom lui répond. 





How far does the fly fly? 
20010807 

Harold pose la question :
6 MPH 4 MPH
Rachel  Eli
10 Miles apart
The fly is on Rachels handlebars. The fly is scared so it flys back and forth at 20 MP H. How far has the fly flown when Rachel and Eli meet? f Penny Nom lui répond. 





Velocity of a pendulum 
20000828 

Mekca pose la question : A pendulum hangs from the ceiling. as the pendulum swings, its distance,d cm, form one wall of the room depends on the number of seconds,t, since it was set in motion. assume that the equation for d as a function of t is: d=80+30cos3.14/3t, t>0. estimate the instantaneous rate of change of d at t=5 by finding the average rates for t=5 to 5.1, t=5 to 5.01, and t=5 to 5.001. Harley Weston lui répond. 





Play ball 
20000203 

Jessie pose la question : Here's a calc question that is probably a lot easier than I am making it. If you have a legendary "baseball problem" for the related rates section of Calc I, and you are given that the runner is running from 2nd to 3rd base at a given rate, and the umpire is standing at home plate, and you are given the distance between the bases on the field, how do you find the rate of change of the angle between the third base line (from the point of the umpire) and the runner? Here is a sample prob: Runner is moving from 2nd to 3rd base at a rate of 24 feet per second. Distance between the bases is 90 feet. What is the rate of change for the angle (theta, as described previously) when the runner is 30 feet from 3rd base? Harley Weston lui répond. 





The average rate of change of a function 
19990420 

Tammy pose la question : Suppose that the average rate of change of a function f over the interval from x=3 to x=3+h is given by 5e^h4cos(2h). what is f'(3)? I would appreciate any help with this question. Harley Weston lui répond. 

