A cone-shaped coffee filter of radius 6 cm and depth 10cm contains water, which drips

With time, t, in minutes, the temperature, H, in degrees Celsius, of a bottle of water put in the refrigerator at t = 0 is given by H = 4 + 16e 0.02t . How fast is the water cooling initially? After 10 minutes? Give units.

According to the US Census, the world population P, in billions, was approximately P = 6.7e 0.011t , where t is in years since January 1, 2007. At what rate was the worlds population increasing on that date? Give your answer in millions of people per year. [The world population growth rate has actually decreased since 2007.]

The power, P, dissipated when a 9-volt battery is put across a resistance of R ohms is given by P = 81 R . What is the rate of change of power with respect to resistance?

With length, l, in meters, the period T , in seconds, of a pendulum is given by T = 2 l 9.8 . (a) How fast does the period increase as l increases? (b) Does this rate of change increase or decrease as l increases?

A plane is climbing at 500 feet per minute, and the air temperature outside the plane is falling at 2C per 1000 feet. What is the rate of change (as a function of time) of the air temperature just outside the plane?

Atmospheric pressure decays exponentially as altitude increases. With pressure, P, in inches of mercury and altitude, h, in feet above sea level, we have P = 30e3.23105h. (a) At what altitude is the atmospheric pressure 25 inches of mercury? (b) A glider measures the pressure to be 25 inches of mercury and experiences a pressure increase of 0.1 inches of mercury per minute. At what rate is it changing altitude?

The gravitational force, F, on a rocket at a distance, r, from the center of the earth is given by F = k r2 , where k = 1013 newton km2. When the rocket is 104 km from the center of the earth, it is moving away at 0.2 km/sec. How fast is the gravitational force changing at that moment? Give units. (A newton is a unit of force.)

A voltage V across a resistance R generates a current I = V R . A constant voltage of 9 volts is put across a resistance that is increasing at a rate of 0.2 ohms per second when the resistance is 5 ohms. At what rate is the current changing?

If is the angle between a line through the origin and the positive x-axis, the area, in cm2, of part of a rose petal is A = 9 16 (4 sin(4)). If the angle is increasing at a rate of 0.2 radians per minute, at what rate is the area changing when = /4?

The potential, , of a charge distribution at a point on the positive x-axis is given, for x in centimeters, by = 2 x2 + 4 x  . A particle at x = 3 is moving to the left at a rate of 0.2 cm/sec. At what rate is its potential changing?

The average cost per item, C, in dollars, of manufacturing a quantity q of cell phones is given by C = a q + b where a, b are positive constants. (a) Find the rate of change of C as q increases. What are its units? (b) If production increases at a rate of 100 cell phones per week, how fast is the average cost changing? Is the average cost increasing or decreasing?

If x2 + y2 = 25 and dx/dt = 6, find dy/dt when y is positive and (a) x = 0 (b) x = 3 (c) x = 4

A pyramid has height h and a square base with side x. The volume of a pyramid is V = 1 3 x2h. If the height remains fixed and the side of the base is decreasing by 0.002 meter/yr, what rate is the volume decreasing when the height is 120 meters and the width is 150 meters?

A thin uniform rod of length l cm and a small particle lie on a line separated by a distance of a cm. If K is a positive constant and F is measured in newtons, the gravitational force between them is F = K a(a + l) . (a) If a is increasing at the rate 2 cm/min when a = 15 and l = 5, how fast is F decreasing? (b) If l is decreasing at the rate 2 cm/min when a = 15 and l = 5, how fast is F increasing?

The Dubois formula relates a persons surface area, s, in meters2, to weight, w, in kg, and height, h, in cm, by s = 0.01w0.25h0.75. (a) What is the surface area of a person who weighs 60 kg and is 150 cm tall? (b) The person in part (a) stays constant height but increases in weight by 0.5 kg/year. At what rate is his surface area increasing when his weight is 62 kg?

A rectangle has one side of 10 cm. How fast is the area of the rectangle changing at the instant when the other side is 12 cm and increasing at 3 cm per minute?

A rectangle has one side of 8 cm. How fast is the diagonal of the rectangle changing at the instant when the other side is 6 cm and increasing at 3 cm per minute?

A right triangle has one leg of 7 cm. How fast is its area changing at the instant that the other leg has length 10 cm and is decreasing at 2 cm per second?

The area, A, of a square is increasing at 3 cm2 per minute. How fast is the side length of the square changing when A = 576 cm2 ?

If two electrical resistances, R1 and R2, are connected in parallel, their combined resistance, R, is given by 1 R = 1 R1 + 1 R2 . Suppose R1 is held constant at 10 ohms, and that R2 is increasing at 2 ohms per minute when R2 is 20 ohms. How fast is R changing at that moment?

A dose, D, of a drug causes a temperature change, T , in a patient. For C a positive constant, T is given by T = C 2 D 3  D2. (a) What is the rate of change of temperature change with respect to dose? (b) For what doses does the temperature change increase as the dose increases?

An item costs $500 at time t = 0 and costs $P in year t. When inflation is r% per year, the price is given by P = 500e rt/100. (a) If r is a constant, at what rate is the price rising (in dollars per year) (i) Initially? (ii) After 2 years? (b) Now suppose that r is increasing by 0.3 per year when r = 4 and t = 2. At what rate (dollars per year) is the price increasing at that time?

For positive constants A and B, the force, F, between two atoms in a molecule at a distance r apart is given by F = A r2 + B r3 . (a) How fast does force change as r increases? What type of units does it have? (b) If at some time t the distance is changing at a rate k, at what rate is the force changing with time? What type of units does this rate of change have?

For positive constants k and g, the velocity, v, of a particle of mass m at time t is given by v = mg k 1 e kt/m . At what rate is the velocity changing at time 0? At t = 1? What do your answers tell you about the motion?

A 10 m ladder leans against a vertical wall and the bottom of the ladder slides away from the wall at a rate of 0.5 m/sec. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is (a) 4 m from the wall? (b) 8 m from the wall?

Gasoline is pouring into a vertical cylindrical tank of radius 3 feet. When the depth of the gasoline is 4 feet, the depth is increasing at 0.2 ft/sec. How fast is the volume of gasoline changing at that instant?

Water is being pumped into a vertical cylinder of radius 5 meters and height 20 meters at a rate of 3 meters3/min. How fast is the water level rising when the cylinder is half full?

A spherical snowball is melting. Its radius is decreasing at 0.2 cm per hour when the radius is 15 cm. How fast is its volume decreasing at that time?

The radius of a spherical balloon is increasing by 2 cm/sec. At what rate is air being blown into the balloon at the moment when the radius is 10 cm? Give units in your answer.

The circulation time of a mammal (that is, the average time it takes for all the blood in the body to circulate once and return to the heart) is proportional to the fourth root of the body mass of the mammal. The constant of proportionality is 17.40 if circulation time is in seconds and body mass is in kilograms. The body mass of a growing child is 45 kg and is increasing at a rate of 0.1 kg/month. What is the rate of change of the circulation time of the child?

A certain quantity of gas occupies a volume of 20 cm3 at a pressure of 1 atmosphere. The gas expands without the addition of heat, so, for some constant k, its pressure, P, and volume, V , satisfy the relation P V 1.4 = k. (a) Find the rate of change of pressure with volume. Give units. (b) The volume is increasing at 2 cm3/min when the volume is 30 cm3. At that moment, is the pressure increasing or decreasing? How fast? Give units.

The metal frame of a rectangular box has a square base. The horizontal rods in the base are made out of one metal and the vertical rods out of a different metal. If the horizontal rods expand at a rate of 0.001 cm/hr and the vertical rods expand at a rate of 0.002 cm/hr, at what rate is the volume of the box expanding when the base has an area of 9 cm2 and the volume is 180 cm3?

A ruptured oil tanker causes a circular oil slick on the surface of the ocean. When its radius is 150 meters, the radius of the slick is expanding by 0.1 meter/minute and its thickness is 0.02 meter. At that moment: (a) How fast is the area of the slick expanding? (b) The circular slick has the same thickness everywhere, and the volume of oil spilled remains fixed. How fast is the thickness of the slick decreasing?

A potter forms a piece of clay into a cylinder. As he rolls it, the length, L, of the cylinder increases and the radius, r, decreases. If the length of the cylinder is increasing at 0.1 cm per second, find the rate at which the radius is changing when the radius is 1 cm and the length is 5 cm.

A cone-shaped coffee filter of radius 6 cm and depth 10 cm contains water, which drips out through a hole at the bottom at a constant rate of 1.5 cm3 per second. (a) If the filter starts out full, how long does it take to empty? (b) Find the volume of water in the filter when the depth of the water is h cm. (c) How fast is the water level falling when the depth is 8 cm?

Water is being poured into the cone-shaped container in Figure 4.86. When the depth of the water is 2.5 in, it is increasing at 3 in/min. At that time, how fast is the surface area, A, that is covered by water increasing? [Hint: A = rs, where r, s are as shown.]

Grit, which is spread on roads in winter, is stored in mounds which are the shape of a cone. As grit is added to the top of a mound at 2 cubic meters per minute, the angle between the slant side of the cone and the vertical remains 45. How fast is the height of the mound increasing when it is half a meter high? [Hint: Volume V = r2h/3, where r is radius and h is height.]

A gas station stands at the intersection of a north-south road and an east-west road. A police car is traveling toward the gas station from the east, chasing a stolen truck which is traveling north away from the gas station. The speed of the police car is 100 mph at the moment it is 3 miles from the gas station. At the same time, the truck is 4 miles from the gas station going 80 mph. At this moment: (a) Is the distance between the car and truck increasing or decreasing? How fast? (Distance is measured along a straight line joining the car and the truck.) (b) How does your answer change if the truck is going 70 mph instead of 80 mph?

The London Eye is a large Ferris wheel that has diameter 135 meters and revolves continuously. Passengers enter the cabins at the bottom of the wheel and complete one revolution in about 27 minutes. One minute into the ride a passenger is rising at 0.06 meters per second. How fast is the horizontal motion of the passenger at that moment?

Point P moves around the unit circle.10 (See Figure 4.87.) The angle , in radians, changes with time as shown in Figure 4.88. (a) Estimate the coordinates of P when t = 2. (b) When t = 2, approximately how fast is the point P moving in the x-direction? In the y-direction?

Figure 4.89 shows the number of gallons, G, of gasoline used on a trip of M miles. (a) The function f is linear on each of the intervals 0 <M< 70 and 70 <M< 100. What is the slope of these lines? What are the units of these slopes? (b) What is gas consumption (in miles per gallon) during the first 70 miles of this trip? During the next 30 miles? (c) Figure 4.90 shows distance traveled, M (in miles), as a function of time t, in hours since the start of the trip. Describe this trip in words. Give a possible explanation for what happens one hour into the trip. What do your answers to part (b) tell you about the trip? (d) If we let G = k(t) = f(h(t)), estimate k(0.5) and interpret your answer in terms of the trip. (e) Find k (0.5) and k (1.5). Give units and interpret your answers.

On February 16, 2007, paraglider Eva Wisnierska11 was caught in a freak thunderstorm over Australia and carried upward at a speed of about 3000 ft/min. Table 4.4 gives the temperature at various heights. Approximately how fast (in F/ per minute) was her temperature decreasing when she was at 4000 feet?

Coroners estimate time of death using the rule of thumb that a body cools about 2F during the first hour after death and about 1F for each additional hour. Assuming an air temperature of 68F and a living body temperature of 98.6F, the temperature T (t) in F of a body at a time t hours since death is given by T (t) = 68 + 30.6e kt. (a) For what value of k will the body cool by 2F in the first hour? (b) Using the value of k found in part (a), after how many hours will the temperature of the body be decreasing at a rate of 1F per hour? (c) Using the value of k found in part (a), show that, 24 hours after death, the coroners rule of thumb gives approximately the same temperature as the formula.

A train is traveling at 0.8 km/min along a long straight track, moving in the direction shown in Figure 4.91. A movie camera, 0.5 km away from the track, is focused on the train. (a) Express z, the distance between the camera and the train, as a function of x. (b) How fast is the distance from the camera to the train changing when the train is 1 km from the camera? Give units. (c) How fast is the camera rotating (in radians/min) at the moment when the train is 1 km from the camera?

A lighthouse is 2 km from the long, straight coastline shown in Figure 4.92. Find the rate of change of the distance of the spot of light from the point O with respect to the angle .

When the growth of a spherical cell depends on the flow of nutrients through the surface, it is reasonable to assume that the growth rate, dV /dt, is proportional to the surface area, S. Assume that for a particular cell dV /dt = 1 3S. At what rate is its radius r increasing?

The length of each side of a cube is increased at a constant rate. Which is greater, the relative rate of change of the volume of the cube, 1 V dV dt , or the relative change of the surface area of the cube, 1 A dA dt ?

A circular region is irrigated by a 20 meter long pipe, fixed at one end and rotating horizontally, spraying water. One rotation takes 5 minutes. A road passes 30 meters from the edge of the circular area. See Figure 4.93. (a) How fast is the end of the pipe, P, moving? (b) How fast is the distance P Q changing when is /2? When is 0?

A water tank is in the shape of an inverted cone with depth 10 meters and top radius 8 meters. Water is flowing into the tank at 0.1 cubic meters/min but leaking out at a rate of 0.004h2 cubic meters/min, where h is the depth of the water in meters. Can the tank ever overflow?

For the amusement of the guests, some hotels have elevators on the outside of the building. One such hotel is 300 feet high. You are standing by a window 100 feet above the ground and 150 feet away from the hotel, and the elevator descends at a constant speed of 30 ft/sec, starting at time t = 0, where t is time in seconds. Let be the angle between the line of your horizon and your line of sight to the elevator. (See Figure 4.94.) (a) Find a formula for h(t), the elevators height above the ground as it descends from the top of the hotel. (b) Using your answer to part (a), express as a function of time t and find the rate of change of with respect to t. (c) The rate of change of is a measure of how fast the elevator appears to you to be moving. At what height is the elevator when it appears to be moving fastest?

In a romantic relationship between Angela and Brian, who are unsuited for each other, a(t)represents the affection Angela has for Brian at time t days after they meet, while b(t) represents the affection Brian has for Angela at time t. If a(t) > 0 then Angela likes Brian; if a(t) < 0 then Angela dislikes Brian; if a(t)=0 then Angela neither likes nor dislikes Brian. Their affection for each other is given by the relation (a(t))2 + (b(t))2 = c, where c is a constant. (a) Show that a(t) a (t) = b(t) b (t). (b) At any time during their relationship, the rate per day at which Brians affection for Angela changes is b (t) = a(t). Explain what this means if Angela (i) Likes Brian, (ii) Dislikes Brian. (c) Use parts (a) and (b) to show that a (t) = b(t). Explain what this means if Brian (i) Likes Angela, (ii) Dislikes Angela. (d) If a(0) = 1 and b(0) = 1 who first dislikes the other?

In a 19th century sea-battle, the number of ships on each side remaining t hours after the start are given by x(t) and y(t). If the ships are equally equipped, the relation between them is (x(t))2 (y(t))2 = c, where c is a positive constant. The battle ends when one side has no ships remaining. (a) If, at the start of the battle, 50 ships on one side oppose 40 ships on the other, what is the value of c? (b) If y(3) = 16, what is x(3)? What does this represent in terms of the battle? (c) There is a time T when y(T )=0. What does this T represent in terms of the battle? (d) At the end of the battle, how many ships remain on the victorious side? (e) At any time during the battle, the rate per hour at which y loses ships is directly proportional to the number of x ships, with constant of proportionality k. Write an equation that represents this. Is k positive or negative? (f) Show that the rate per hour at which x loses ships is directly proportional to the number of y ships, with constant of proportionality k. (g) Three hours after the start of the battle, x is losing ships at the rate of 32 ships per hour. What is k? At what rate is y losing ships at this time?

In Problems 5354, explain what is wrong with the statement If the radius, R, of a circle increases at a constant rate, its diameter, D, increases at the same constant rate.

In Problems 5354, explain what is wrong with the statement If two variables x and y are functions of t and are related by the equation y = 1 x2 then dy/dt = 2x.

In Problems 5556, give an example of: Two functions f and g where y = f(x) and x = g(t) such that dy/dt and dx/dt are both constant.

In Problems 5556, give an example of: Two functions g and f where x = g(t) and y = f(x) such that dx/dt is constant and dy/dt is not constant.

Are the statements in Problems 5758 true of false? Give an explanation for your answer. If the radius of a circle is increasing at a constant rate, then so is the circumference.

Are the statements in Problems 5758 true of false? Give an explanation for your answer. If the radius of a circle is increasing at a constant rate, then so is the area.

The light in the lighthouse in Figure 4.95 rotates at 2 revolutions per minute. To calculate the speed at which the spot of light moves along the shore, it is best to differentiate: (a) r2 = 52 + x2 (b) x = r sin (c) x = 5 tan (d) r2 = 22 + x2