Related rates Definition and 361 Threads

Homework Statement When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV^1.14=C, where C is a constant. Suppose that at a certain instant the volume is 600 cm^3 and the pressure is 80 kPa and is decreasing at a rate of 10...

Homework Statement A street light is at the top of a 18 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 30 ft from the base of the pole? Homework Equations...

Homework Statement A spotlight on theground shines on a wall 12 m away. If a man 2m tall walks from the spotlight toward the building at a speed of 1.6m/s. how fast is the length of his shadow on the building decreasing when he is 4m from the building? The Attempt at a Solution x=4m...

Homework Statement http://www.kent.k12.wa.us/pcpow/questions/calc/0401trickortreat/index.html This is an old problem, and I have been able to solve one and two, I believe the answers are 1. pi cm/second 2. 50(pi) cm^3/second 3. This is the one I am not sure of. How can I solve...

Homework Statement A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is π/3, this angle is decreasing at a rate of π/6 rad/min. How fast is the plane traveling at that time? The Attempt at a Solution I...

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 The Attempt at a Solution I have done the pythagreon...

Homework Statement a spotlight won the ground shines on a wall 12 m away if am man 2 m tall walks from the spotlight towards the building at a speed of 1.6 m/s how fast is the length of his shadow on the building decreasing when he si 4 m from the building? Homework Equations using...

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 7 inches deep? I...

Homework Statement Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. The height of the pile is increasing at a rate of ____ feet per minute when the pile is...

Homework Statement A satellite is in an orbit around earth. The distance from the center of the Earth is described by r= 4995/(1+.12cos@) R earth= 3960 mi find the rate at which the altitude is changing at the instant where @=120 degrees. d@/dt= 2.7 degrees/min 2. Notes...

Conical water tank, vertex down. Radius of 10ft, 24 ft high. Water flows into tank at 20 cubic ft/ min. How fast is the depth of the water increasing at 16ft? r=10ft y=16ft dV/dt=20 cubic ft/min dy/dt=? So here was my attempt: V=1/3 (pi r^2) y I treated r as a constant so I could...

Homework Statement "An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of the plane is 600 miles per hour. Find the rate at which the angle of elevation \theta is changing when the angle is 30\circ" variables: x = ground distance of plane from the...

Homework Statement A light is hung 15ft above a straight horizontal path. If a man 6 ft tall is walking away from the light at the rate of 5ft/sec. How fast is his shadow lengthening? And at what rate is the tip of the man's shadow moving? y = 6? or 15? dx/dt = 6 Homework Equations...

Homework Statement a light shines from the op of a pole 50ft high a ball is dropped from the same height from a point 30 feet away from the light , how fast is the shadow of the ball moving along the ground .5second later. So is the ball falling at 32ft/s^2 would we set it up...

Homework Statement An airplane flying horizontally at an altitude of 3 miles and at a speed of 480mi/hr passes directly above an observer on the ground. How fast is the distance from the observer to the airplane increasing 30 seconds later? Homework Equations The Attempt at a...

Homework Statement A man walks along a straight path at 5km/hr. A search light is located on the ground 30 metres form the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 20 metres from the point on the path closest to the searchlight...

Homework Statement A Missile rises vertically from a point on the ground 75,000 feet from a radar station. If the missle is rising at the rate of 16,500 feet per minute at the instant when it is 38,000 feet high. What is the rate of change, in radians per minute, of the missile's angle of...

a) 1. Homework Statement A water trough is 10m long, and a cross section has the shape of an isosceles triangle that is 1m across at the top and 50cm high. The trough is being filled with water at a rate of 0.4m^3/min. How fast is the water level rising when the water is 40cm deep? b) As a...

Homework Statement A water tank has the shape of an inverted circular cone with a base diameter of 8m and a height of 12m. a) If the tank is being filled with water at the rate of 5m^3/min, at what rate is the water level increasing when the water is 5m deep? b) If the tank is full of water...

1.Suppose the wood nymphs and satyrs are having a hot party in honor of Bacchus and the wine is flowing freely from the bottom of a giant cone-shaped barrel which is 12 feet deep and 6 feet in radius at the top. if the wine is disappearing at a rate of 6 cubic feet per hour, at what rate is the...

There is a trough that is 10 meters long, 6 meters wide, and is in the shape of an equalateral triangle. The volume is changing at a rate of 0.2 m/s a second. The goal is to find the rate at which the height is changing at 2m. So I initially set the problem up as V=0.5bhl (where b=width...

Homework Statement This is the problem I am having... All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter (b) 30 centimeters Homework Equations The equation I am using is V = S³ The...

Homework Statement When the area of an expanding square, in square units, is increasing three times as fast as its side is increasing, in linear units, the side is a.) 2/3 b.) 3/2 c) 3 d) 2 e) 1 Homework Equations A = s^2 dA/dt = 3s^2 The Attempt at a Solution Can...

Homework Statement Mr. Wilson is standing near the top of a ladder 24 feet long which is leaning against a vertical wall of his house. Dennis the little boy next door, ties a rope from his tricycle to the bottom of the ladder and starts to pull the foot of the ladder away from the house...

Homework Statement A wight W is attached to a rope 50 ft long that passes over a pulley at point P, 20 ft above the ground. The other end of the rope is attached to a truck at a point A, 2 ft above the round. If th truck moves away at the rate of 9 ft/s, how fast is the weight rising when it...

Homework Statement a snowball is rolling donw a hill, its radius is changing at a rate of 2 cm/min. what is the rate of change of the area, when the radius is 8 cm? Homework Equations da/dt= 2pir2dr/dt dat/dt=2pi(8)(2) 32pi ? is that right The Attempt at a Solution 32pi

Homework Statement Let A be the area of the region in the first quadrant under the graph of y = cos (x) and above the line y = k for 0 <= k <= 1. a.) Determine A in terms of k. b) Determine the value of A when k = 1/2. c) If the line y = k is moving upward at the rate of ( 1 / pi )...

I believe this is a related rates problem. I attempted part 2 but I'm not sure about the equation for the maximum height. is it the derivative ?

Please Help! Related Rates Satellite Problem Homework Statement A satellite is in an elliptical orbit around the earth. When the satellite is located at any point P on this elliptical orbit, the distance r (in miles) from the center of the Earth at point E to Point P is given by the...

Hopefully I posted this in the right place. The setup is standard - A ladder 10 ft long rests against a vertical wall. The bottom of the ladder slides away from the wall at a rate of 1 ft/sec. If we let y be the vertical distance and x the horizontal distance, we use the Pythagorean theorem...

Related rates prob. [solved] A hemispherical bowl of radius 8 in. is being filled with water at a constant rate. If the water level is rising at the rate of 1/3 in./s at the instant when the water is 6 in. deep, find how fast the water is flowing in by using the fact that if V is the volume of...

Homework Statement The radius of a circle is increasing at a nonzero rate, and at a certain instant, the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference The Attempt at a Solution c=circumference a=area If the rate of change...

Homework Statement A plane is approaching an observer from an altitude of 5 mi at a dx/dt of 600 mi/h. Find the d(theta)/dt when theta is 30 degrees, 60 degrees, and 75 degrees. Homework Equations tan(theta) = x/y 5csc(theta) = r The Attempt at a Solution For my d(theta)/dt I...

This due today! Water flows from a tank of constant cross-sectional area 56 ft2 through an orifice of constant cross-sectional area 1.5 ft2 located at the bottom of the tank. Initially the height of the water in the tank was 20 and its height t sec later is given by the following equation...

Homework Statement A man 6 feet tall walks as a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light, a) At what rate is the tip of his shadow moving? b) At what rate is the length of his shadow changing? Homework...

Homework Statement Car A is going north, car B is going west, each are approaching an intersection on their respective highways. At an instant, car A is .3km from its intersection while car B is .4 km from it's intersection. Car A travels at 90km/h while car B travels 80km/h. Find the rate at...

Homework Statement A rectangular prism has its length increasing by 12 cm/min, its width increasing by 4 cm/min and its height increasing by 2 cm/min. How fast is it's volume changing when the dimensions are 200 cm in length, 50 cm in width and 30 cm in height? Homework Equations...

this assignment is on related rates. I believe questions 3/4 are on linear approximation and relate back to the last unit. My problem is that I am entirely unconfident on this work, and am going to be taking a test on the material soon. If someone could check my work, I'd be delighted...

Homework Statement A rocket is launched vertically and is tracked by a radar station located on the ground 5 mi from the launch pad. Suppose that the elevation angle θ of the line of sight to the rocket is increasing at 3° per second when θ=60°. What is the velocity of the rocket at this...

A man standing 3 feet from the base of a lamppost casts a shadow 4 feet long. If the man is 6 feet tall and walks away from the lamppost at a speed of 400 feet per minute, at what rate will his shadow lengthen? How fast is the tip of his shadow moving? I'm unsure of how to solve the 2nd part, a...

Homework Statement A rectangular prism has its length increasing by 12 cm/min, its width increasing by 4 cm/min and its height increasing by 2 cm/min. How fast is it's volume changing when the dimensions are 200 cm in length, 50 cm in width and 30 cm in height? Homework Equations...

Homework Statement A balloon is rising at a constant speed of 5 ft/s. A boy is cycling along a straight road at a speed of 15 ft/s. When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 seconds later. Homework...

Homework Statement A swimming pool is 24 m long by 8 m wide, 1 m deep at the shallow end and 3 m deep at the deep end, the bottom being an inclined plane. If water is pumped into the empty pool at a rate of 2m^3/min, then how fast is the water level rising at the moment when the water is 1 m...

Even problem, very please! A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s. a) At what rate is his distance from second base decreasing when he is halfway to the first base? b) At what rate is his distance from third...

[SOLVED] Related Rates Homework Statement Two aircraft are in the vicinity of a control center. Both are at the same altitude. Plane 1 is 36 nautical miles from the center and approaching it at a rate of 410 knots. Plane 2 is 41 nautical miles from the center and approaching it at a rate of...

Homework Statement How fast is the area of a square increasing when the side is 3 meters in length and growing at a rate of 0.8meter/minute? Homework Equations SA=LW L=3 dL/dt=0.8 The Attempt at a Solution I have no clue how to start this and I'm bad at word problems.

Homework Statement A rectangular swimming pool 16m x 10m is being filled at a rate of .8m^3/min. How fast is height of the water rising? Homework Equations V = LWH The Attempt at a Solution H = V/LW (dh/dt) = LW(dv/dt) (dh/dt) = (10)(16)(.8) This question looks so easy...

[SOLVED] Spherical Balloon - Related Rates Problem Homework Statement A spherical balloon is inflated so that its radius increases at a rate of 1 cm/min. How fast is the volume increasing when: a) the diameter is 2000 cm b) the surface area is 324 pi cm^2 ---> I have solved this already...

A sailor in a boat 10 km off a straight coastline wants to reach a point 20km along the coast in the shortest possible time. Toward what point on the shore should he head if he can row at 4 km/h and run at 12 km/h? i believe this involves optomizeation as well ralated rates but i am unsure as...

Need help! Related rates! Homework Statement I am stuck somewhere on a related rates problem, i think that i am missing something rather obvious, but i cannot figure out so far. -A street light is mounted at the top of the 15 ft-tall pole. A man 6 ft tall walks away from the pole with a...