Related rates Definition and 371 Threads

Homework Statement Air is being pumped into a spherical balloon. Suppose we know the surface area of the balloon increases at a rate of 20cm^2/s when its radius is 4cm. What is the rate its volume is changing at that instant?Homework Equations The Attempt at a Solution I went... dv/dt = 4∏...

Homework Statement A circular ring of wire of radius r0 lies in a plane perpendicular to the x-axis and is centered at the origin. The ring has a positive electric charge spread uniformly over it. The electric field in the x-axis direction, E, at the point given by E=kx/((x^2 +r0^2)^(3/2))...

Homework Statement A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks 20 miles apart, the concentration of the combined deposits on the line joining them, at a distance x from one stack, is...

I've been trying to figure out where my mistake lies in the first solution. Some help would be appreciated. I did notice I got the same solution twice, so I assume I just calculated dr/dt twice and I need to use a different equation for dh/dt? Is dh/dt=(3/4)*dr/dt? 1. Sand falls from a...

I need help with this problem in my calculus book. An airplane at an altitude of 10,000 feet is flying at a constant speed on a line which will take it directly over an observer on the ground. If, at a given instant, the observer notes that the angle of elevation of the airplane is 60 degrees...

Homework Statement A highway patrol plane is flying 1 mile above a long, straight road, with constant ground speed of 120 m.p.h. Using radar, the pilot detects a car whose distance from the plane is 1.5 miles and decreasing at a rate of 136 m.p.h. How fast is the car traveling along the...

1. At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/hr and ship B is sailing north at 25 km/hr. How fast is the distance between the ships changing at 4:00 pm? 2. None 3. I have the distance as 150 km. I have the variables \frac{dx}{dt} = 35 and...

Hi all! I was looking through my old calculus book and was looking at a problem on related rates. I looked at the problem that was in the related rates section and I saw what was an easier way of doing it. The problem was this: Car A is traveling west at 50 mi/hr and Car B is traveling north...

Homework Statement A snowball melts such that the volume decreases at a rate of 1cm3/min. At what rate is the diameter decreasing when diameter=10? I know the answer is \frac{-1cm}{50∏ per ?}. My problem is with the units on the bottom. It was given as seconds, but shouldn't it be per minute...

A child loves to watch as you fill a transparent plastic bottle with shampoo. Every horizontal cross section of the bottle is circular, but the diameters of the circles have different values. You pour the brightly colored shampoo into the bottle at a constant rate of 17.0 cm3/s. At what rate...

Homework Statement A lake is polluted by water from a plant located on its shore. Ecologist determine that when the level of pollutant is x parts per million (ppm), there will be F fish in the lake. When there are 4,000 fish in the lake, the pollution is increasing at a rate of 1.4ppm/year. At...

If 1200∏ cm^2 of material is available to make a cylindrical can with a circular base an open top, find the largest possible volume of the can. the formulas i used: v=∏* r^2 * h surface area = 2∏r^2 + 2rh my attempt: 1200=r^2h∏ h=1200/r^2∏ SA=2∏r^2+2∏r(1200/∏r^2) =2∏r^2+...

Homework Statement A solution is being poured into a conical filter in a chemistry experiment at a rate of 5cm^3/min. The filter is 15 cm high with a diameter of 10 cm at the top. The solution is dropping out of the filter at a rate of 1 cm^3/min. Determine the rate at which the height of...

Homework Statement "At a given instant the legs of a right triangle are 8in. and 6in., respectively. The first leg decreases at 1in/min and the second increases at 2in/min. At what rate is the area increasing after 2 minutes?"Homework Equations A=\frac{1}{2}bh \frac{db}{dt}=-1...

I am assuming that I am setting this up incorrectly or substituting the incorrect values. A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with...

Homework Statement A container of constant volume contains a quantity of gas under pressure. At t=0, the pressure is 4 psi and the temperature is 15°C per minute. What is the rate of change of the pressure inside the container at time t=0? Homework Equations \frac{P}{T} = constant The...

Homework Statement 12.) A trough is 15ft long and 4ft across the top as shown in the figure. Its ends are isosceles triangles with height 3ft. Water runs into the trough at the rate of 2.5 ft3/min. How fast is the water level rising when it is 2 ft deep?Homework Equations The Attempt at a...

Homework Statement Apostol, Vol 1: Section 4.12 Problem 26 Water flows into a hemispherical tank of radius 10 feet (flat side up). At any instant, let h denote the depth of the water, measured from the bottom, r the radius of the surface of the water, and V the volume of the water in the...

Hi Guys, I have a general question (not necessarily a homework question) about the concept of related rates in differential calculus. Most related rates problems present to you a question that generally asks, After x time has elapsed, or At t= __, what is the rate of change between...

Homework Statement A swimming pool is 50m long and 20m wide. Its depth decreases linearly along the length from 3m to 1m. It is initially empty and is filled at a rate of 1m^3/min. How fast is the water level rising 90 minutes after the filling began? The Attempt at a Solution I'm...

Homework Statement Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown? Homework Equations AB/AD = BC/DE The Attempt at a Solution I've been trying to do this problem. I looked online and saw an explanation without directly...

Theres two parts to the question. Each part also has two parts to it. I am pretty sure my work i did so far is correct( double check please) but I am not sure about the bolded parts A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat.? a) The winch pulls...

The problem is: For the baseball diamond in exercise 33, suppose the player is running from first to second at a speed of 28 feet per second. Find the rate at which the distance from home plate is changing when the player is 30 feet from second base. (the picture basically shows a square with 90...

Homework Statement A water tank has the shape of an inverted right-circular cone, with radius at the top 15 meters and depth 12 meters. Water is flowing into the tank at rate of 2 cublic meters per minute. How fast is the depth of water in the tank increasing at the instant when the depth is 8...

Homework Statement In the special theory of relativity the mass of a particle moving at speed v is given by the expression m/(1-(v2/c2)) where m is the mass at rest and c is the speed of light. At what rate is the mass of the particle changing when the speed of the particle is (1/2)c and is...

Homework Statement A 16-ft ladder is sliding down a wall at a rate of 4 ft/sec. Find the velocity of the top of the ladder at t=s if the bottom is 5ft from the wall at t=0. Homework Equations It's a related rates problem. Not sure how I'm supposed to incorporate t=s into the problem, or if...

Related Rates Verification, Help Please! Homework Statement The capacity of Bob's truck is 72 lbs. He needs to carry as much tar paper as possible and 1 square yard weighs 5 lbs. If the paper comes off the roll at 2 square feet per second, how long will it take to fill his truck? Homework...

Homework Statement A particle is moving along the ellipse x2/16 + y2/4 = 1. At each time t its x and y coordinates are given by x = 4cost, y = 2sint. At what rate is the particle's distance from the origin changing at time t? At what rate is the distance from the origin changing when t = pi/4...

Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley P (see the figure). The point Q is on the floor h = 12 ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2.5 ft/s. How fast is cart B moving toward Q at the instant when...

A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.1 ft/s, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 8 ft from the wall?

Homework Statement Sand falls from a conveyor belt at the rate of 10 m^3/min onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the a) height and b) radius changing when the pile is 4m high? Homework Equations h = 3/8 * 2r...

Homework Statement On February 16, 2007, paraglider Eva Wisnierska was caught in a freak thunderstorm over Australia and carried upward at a speed of about 3000 ft/min. The table below gives the temperature at various heights. Approximately how fast (in ◦F/min) was her ambient temperature...

Homework Statement At 9 a.m ship A is 50 km [E] of ship B. Ship a is sailing [N] at 40 km/h and ship B is sailing [S] at 30 km/h. How fast is the distance between them changing at 12 am? Homework Equations x^2 + y^2 = z^2 dz/dt [t=3h] = ? db/dt = 30 km/h da/dt = 40 km/h The...

Homework Statement If xy^2 = 12 and dy/dt = 6, find dx/dt when y = 2. Homework Equations The Attempt at a Solution My teacher wants us to follow a five step method for solving related rates: Step 1 [Information]: Assign variable letters to known and unknown quantities xy^2 = 12 dy/dt = 6 dx/dt...

Homework Statement A ship moving at 8 mi per hour, Sails W for 2 hours, then turns N 30 E. A search light, placed at the starting point, follows the ship. Find how fast the light is rotating, (a) 3 hours after the start; (b) just after the turn. Homework Equations The Attempt at...

Homework Statement A hemispherical dome has a diameter of 100m. A search light was placed at point A as shown at the middle of the dome at B. A balloon was released vertically at a velocity of 4m/s. How fast will the shadow of the balloon move alone the roof if it traveled 25 m vertically...

Homework Statement An 18 meter ladder is sliding down a vertical wall at a rate of 2.5 m/s. Find the speed of the lower end of the ladder when it is 12 meters from the wall. Homework Equations Pythagorean Theorem The Attempt at a Solution Let h = height of the wall L = length of the...

Homework Statement An engineering student holds her open compass perpendicular to the drafting board, touching the board with both tips of the compass. She slowly closes the compass so that the tips move toward each other with a speed of 2 v_naught = 0.060 m/s. Initially the angle between...

Could someone verify my answer for this question? 13m ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2m/s. How fast will the foot be moving away from the wall when the top is 5m from the ground? My answer: use pythagoras to set up equation...

Homework Statement Water flows into a cubical tank at a rate of 19 L/s. If the top surface of the water in the tank is rising by 3.7 cm every second, what is the length of each side of the tank? Homework Equations v=L^3 The Attempt at a Solution so what I started doing was...

Here is the question verbatim: The radius of a right-circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as the radius. When the radius is 1 foot, the altitude is 6 feet. When the radius is 6 feet, the volume is...

EDIT: I think I figured it out - sorry for taking up space. I posted my answer below.* Homework Statement A light is at the top of a 16-ft pole. A boy 5 ft tall walks away from the pole at a rate of 4 ft/sec. a) At what rate is the tip of his shadow moving when he is 18 ft from the pole...

Homework Statement A winch at the top of a 12 metre building pulls a pipe of the same length to a vertical positon, as shown in the figure below. The winch pulls the rope at a rate of -0.2 m/s. Find the rate of the vertical change and horiziontal change at the end of the pipe when y=6...

(b]1. Homework Statement [/b] A spherical weather balloon has a radius of 1m when it is 1500m high. You observe that the radius increases at a rate of 2cm/min as it continues to rise. At what rate is the surface area increasing when the radius is 4m? Homework Equations I thought...

I'm not sure whether that's considered physics, algebra or calculus, but it might include all three that I just thought to post it here. Homework Statement http://img824.imageshack.us/img824/924/ripples.jpg A radius of a ripple is increasing at a rate of 6 inches per second...

Homework Statement A man 6 feet tall walks away from a streetlight that is 18 feet tall. If the length of his shadow is changing at a rate of 3 feet per second when he is 25 feet away from the base of the light, how fast is he walking away from the light at this moment? Homework Equations...

Homework Statement A plane flying with a constant speed of 300 km/h passes over a ground radar station at an altitude of 1 km and climbs at an angle of 30 degrees. At what rate is the distance from the plane to the radar station increasing a minute later? Homework Equations a2+b2=c2The...

Homework Statement The measure of one of the acute angles of a right triangle is decreasing at the rate 1/36 pi rad/sec. If the length of the hypotenuse is constant at 40cm, find how fast the area is changing when the measure of the acute angle is 1/6 pi. Homework Equations The...

Homework Statement Sand is pouring from a pipe at the rate of 16 cubic feet per second. If the falling sand forms a conical pile on the ground whose altitude is always 1/4 the diameter of the base, how fast is the altitude increasing when the pile is 4 feet high? Homework Equations V =...

Homework Statement The area A of a triangle with sides of lengths a and b enclosing an angle of measure \theta is: A=1/2 ab sin (\theta) How is dA/dt related to d\theta/dt if side a and side b is constant? Homework Equations The Attempt at a Solution I am pretty sure that I...