Electricity The combined electrical resistance of and

In Exercises 1 4, assume that and are both differentiable functions of and find the required values of and y12x

In Exercises 1 4, assume that and are both differentiable functions of and find the required values of and 12x

In Exercises 1 4, assume that and are both differentiable functions of and find the required values of and 12x4

In Exercises 1 4, assume that and are both differentiable functions of and find the required values of and 2x4y

In Exercises 58, a point is moving along the graph of the given function such that is 2 centimeters per second. Find for the given values of x4y

In Exercises 58, a point is moving along the graph of the given function such that is 2 centimeters per second. Find for the given values of 4y

In Exercises 58, a point is moving along the graph of the given function such that is 2 centimeters per second. Find for the given values of 4y1

In Exercises 58, a point is moving along the graph of the given function such that is 2 centimeters per second. Find for the given values of y12

In Exercises 9 and 10, using the graph of (a) determine whether is positive or negative given that is negative, and (b) determine whether is positive or negative given that is positive. 12x

In Exercises 9 and 10, using the graph of (a) determine whether is positive or negative given that is negative, and (b) determine whether is positive or negative given that is positive. 12x

Consider the linear function If changes at a constant rate, does change at a constant rate? If so, does it change at the same rate as Explain. 12x2

Consider the linear function If changes at a constant rate, does change at a constant rate? If so, does it change at the same rate as Explain. 2x2y

Find the rate of change of the distance between the origin and a moving point on the graph of if centimeters per second. x2y

Find the rate of change of the distance between the origin and a moving point on the graph of if centimeters per second. 2y3

Area The radius of a circle is increasing at a rate of 3 centimeters per minute. Find the rates of change of the area when (a) centimeters and (b) centimeters. 2y3x

Area Let be the area of a circle of radius that is changing with respect to time. If is constant, is constant? Explain. y3x

Area The included angle of the two sides of constant equal length of an isosceles triangle is (a) Show that the area of the triangle is given by (b) If is increasing at the rate of radian per minute, find the rates of change of the area when and (c) Explain why the rate of change of the area of the triangle is not constant even though is constant. 3x2

Volume The radius of a sphere is increasing at a rate of 2 inches per minute. (a) Find the rate of change of the volume when inches and inches. (b) Explain why the rate of change of the volume of the sphere is not constant even though is constant. 3x2

Volume A spherical balloon is inflated with gas at the rate of 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is (a) 30 centimeters and (b) 60 centimeters? x2y

Volume 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 and (b) 10 centimeters? 2y

Surface Area The conditions are the same as in Exercise 20. Determine how fast the surface area is changing when each edge is (a) 1 centimeter and (b) 10 centimeters. 2y

Volume The formula for the volume of a cone is Find the rate of change of the volume if is 2 inches per minute and when (a) inches and (b) inches. yx

Volume At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high? yx

Depth A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep. x3

Depth A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end (see figure). Water is being pumped into the pool at cubic meter per minute, and there is 1 meter of water at the deep end. (a) What percent of the pool is filled? (b) At what rate is the water level rising? x3y

Depth A trough is 12 feet long and 3 feet across the top (see figure). Its ends are isosceles triangles with altitudes of 3 feet. (a) If water is being pumped into the trough at 2 cubic feet per minute, how fast is the water level rising when is 1 foot deep? (b) If the water is rising at a rate of inch per minute when determine the rate at which water is being pumped into the trough. x3y

Moving Ladder A ladder 25 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. (a) How fast is the top of the ladder moving down the wall when its base is 7 feet, 15 feet, and 24 feet from the wall? (b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. (c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall. 3y

Construction A construction worker pulls a five-meter plank up the side of a building under construction by means of a rope tied to one end of the plank (see figure). Assume the opposite end of the plank follows a path perpendicular to the wall of the building and the worker pulls the rope at a rate of 0.15 meter per second. How fast is the end of the plank sliding along the ground when it is 2.5 meters from the wall of the building? 3yx

Construction A winch at the top of a 12-meter building pulls a pipe of the same length to a vertical position, as shown in the figure. The winch pulls in rope at a rate of meter per second. Find the rate of vertical change and the rate of horizontal change at the end of the pipe when yx3

Boating A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat (see figure). (a) The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to the dock? (b) Suppose the boat is moving at a constant rate of 4 feet per second. Determine the speed at which the winch pulls in rope when there is a total of 13 feet of rope out. What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock? x3

Air Traffic Control An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other (see figure). One plane is 150 miles from the point moving at 450 miles per hour. The other plane is 200 miles from the point moving at 600 miles per hour. (a) At what rate is the distance between the planes decreasing? (b) How much time does the air traffic controller have to get one of the planes on a different flight path? x3x

Air Traffic Control An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna (see figure on previous page). When the plane is 10 miles away the radar detects that the distance is changing at a rate of 240 miles per hour. What is the speed of the plane? x3x

Sports A baseball diamond has the shape of a square with sides 90 feet long (see figure). A player running from second base to third base at a speed of 28 feet per second is 30 feet from third base. At what rate is the players distance from home plate changing? 3xy

Sports 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. xy

Shadow Length A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground (see figure). 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? xyx

Shadow Length Repeat Exercise 35 for a man 6 feet tall walking at a rate of 5 feet per second toward a light that is 20 feet above the ground (see figure). yx2

Machine Design The endpoints of a movable rod of length 1 meter have coordinates and (see figure). The position of the end on the axis is where is the time in seconds. (a) Find the time of one complete cycle of the rod. (b) What is the lowest point reached by the end of the rod on the axis? (c) Find the speed of the axis endpoint when the axis endpoint is yx2x

Machine Design Repeat Exercise 37 for a position function of Use the point for part (c). x2x2

Evaporation As a spherical raindrop falls, it reaches a layer of dry air and begins to evaporate at a rate that is proportional to its surface area Show that the radius of the raindrop decreases at a constant rate. x2x2

Electricity The combined electrical resistance of and connected in parallel, is given by where and are measured in ohms. and are increasing at rates of 1 and 1.5 ohms per second, respectively. At what rate is changing when ohms and ohms? 2x21

Adiabatic Expansion When a certain polyatomic gas undergoes adiabatic expansion, its pressure and volume satisfy the equation where is a constant. Find the relationship between the related rates and x21y

Roadway Design Cars on a certain roadway travel on a circular arc of radius In order not to rely on friction alone to overcome the centrifugal force, the road is banked at an angle of magnitude from the horizontal (see figure). The banking angle must satisfy the equation where is the velocity of the cars and feet per second per second is the acceleration due to gravity. Find the relationship between the related rates and 21y

Angle of Elevation A balloon rises at a rate of 3 meters per second from a point on the ground 30 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground. 1yx

Angle of Elevation A fish is reeled in at a rate of 1 foot per second from a point 10 feet above the water (see figure). At what rate is the angle between the line and the water changing when there is a total of 25 feet of line out? 1yxx

Angle of Elevation An airplane flies at an altitude of 5 miles toward a point directly over an observer (see figure). The speed of the plane is 600 miles per hour. Find the rates at which the angle of elevation is changing when the angle is (a) (b) and yxx2

Linear vs. Angular Speed A patrol car is parked 50 feet from a long warehouse (see figure). The revolving light on top of the car turns at a rate of 30 revolutions per minute. How fast is the light beam moving along the wall when the beam makes angles of (a) (b) and (c) with the line perpendicular from the light to the wall? xx2

Linear vs. Angular Speed A wheel of radius 30 centimeters revolves at a rate of 10 revolutions per second. A dot is painted at a point on the rim of the wheel (see figure). (a) Find as a function of (b) Use a graphing utility to graph the function in part (a). (c) When is the absolute value of the rate of change of greatest? When is it least? (d) Find when and xx21

Flight Control An airplane is flying in still air with an airspeed of 240 miles per hour. If it is climbing at an angle of find the rate at which it is gaining altitude. x21xy

Security Camera A security camera is centered 50 feet above a 100-foot hallway (see figure). It is easiest to design the camera with a constant angular rate of rotation, but this results in a variable rate at which the images of the surveillance area are recorded. So, it is desirable to design a system with a variable rate of rotation and a constant rate of movement of the scanning beam along the hallway. Find a model for the variable rate of rotation if feet per second. 21xy

Think About It Describe the relationship between the rate of change of and the rate of change of in each expression. Assume all variables and derivatives are positive. (a) (b) 1xy

Acceleration In Exercises 51 and 52, find the acceleration of the specified object. (Hint: Recall that if a variable is changing at a constant rate, its acceleration is zero.) Find the acceleration of the top of the ladder described in Exercise 27 when the base of the ladder is 7 feet from the wall. 1xy4

Acceleration In Exercises 51 and 52, find the acceleration of the specified object. (Hint: Recall that if a variable is changing at a constant rate, its acceleration is zero.) Find the acceleration of the boat in Exercise 30(a) when there is a total of 13 feet of rope out. xy4

Modeling Data The table shows the numbers (in millions) of single women (never married) and married women in the civilian work force in the United States for the years 1993 through 2001. (Source: U.S. Bureau of Labor Statistics) (a) Use the regression capabilities of a graphing utility to find a model of the form for the data, where is the time in years, with corresponding to 1993. (b) Find Then use the model to estimate for if it is predicted that the number of single women in the work force will increase at the rate of 0.75 million per year. 4x

Moving Shadow A ball is dropped from a height of 20 meters, 12 meters away from the top of a 20-meter lamppost (see figure). The balls shadow, caused by the light at the top of the lamppost, is moving along the level ground. How fast is the shadow moving 1 second after the ball is released? (Submitted by Dennis Gittinger, St. Philips College, San Antonio, TX) 4x2