Solved: Water is leaking out of an inverted conical tank at a rate of 10,000 cm min at

If is the volume of a cube with edge length and the cube expands as time passes, find in terms of

. (a) If is the area of a circle with radius and the circle expands as time passes, find in terms of . (b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of , how fast is the area of the spill increasing when the radius is 30 m?

Each side of a square is increasing at a rate of . At what rate is the area of the square increasing when the area of the square is ?

The length of a rectangle is increasing at a rate of and its width is increasing at a rate of . When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?

A cylindrical tank with radius 5 m is being filled with water at a rate of . How fast is the height of the water increasing?

The radius of a sphere is increasing at a rate of . How fast is the volume increasing when the diameter is 80 mm?

. Suppose , where and are functions of . (a) If , find when . (b) If , find when .

Suppose , where and are functions of . (a) If , find when and . (b) If , find when and .

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A particle is moving along a hyperbola . As it reaches the point , the -coordinate is decreasing at a rate of . How fast is the -coordinate of the point changing at that instant?

1114 (a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time t. (d) Write an equation that relates the quantities. (e) Finish solving the problem.A plane flying horizontally at an altitude of 1 mi and a speed of 500 mih passes directly over a radar station. Find the rate atwhich the distance from the plane to the station is increasing when it is 2 mi away from the station.

1114 (a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time t. (d) Write an equation that relates the quantities. (e) Finish solving the problem.If a snowball melts so that its surface area decreases at a rate of 1 cm min, find the rate at which the diameter decreases when the diameter is 10 cm.

1114 (a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time t. (d) Write an equation that relates the quantities. (e) Finish solving the problem.A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 5 fts along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole?

1114 (a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time t. (d) Write an equation that relates the quantities. (e) Finish solving the problem.. At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 kmh and ship B is sailing north at 25 kmh. How fast is the distance between the ships changing at 4:00 PM?

Two cars start moving from the same point. One travels south at 60 mih and the other travels west at 25 mih. At what rate is the distance between the cars increasing two hours later?

A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 ms, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?

A man starts walking north at 4 fts from a point . Five minutes later a woman starts walking south at 5 fts from a point 500 ft due east of . At what rate are the people moving apart 15 min after the woman starts walking?

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 fts. (a) At what rate is his distance from second base decreasing when he is halfway to first base? (b) At what rate is his distance from third base increasing at the same moment?

The altitude of a triangle is increasing at a rate of 1 cmmin while the area of the triangle is increasing at a rate of 2 cm min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is ?

0. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 ms, how fast is the boat approaching the dock when it is 8 m from the dock?

At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 kmh and ship B is sailing north at 25 kmh. How fast is the distance between the ships changing at 4:00 PM?

A particle moves along the curve . As the particle passes through the point , its -coordinate increases at a rate of . How fast is the distance from the particle to the origin changing at this instant?

Water is leaking out of an inverted conical tank at a rate of 10,000 cm min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cmmin when the height of the water is 2 m, find the rate at which water is being pumped into the tank.

A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 ft min, how fast is the water level rising when the water is 6 inches deep?

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 water at the rate of 0.2 , how fast is the water level rising when the water is 30 cm deep?

A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. A cross-section is shown in the figure. If the pool is being filled at a rate of 0.8 , how fast is the water level rising when the depth at the deepest point is 5 ft?

Gravel is being dumped from a conveyor belt at a rate of 30 , and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

A kite 100 ft above the ground moves horizontally at a speed of 8 fts. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?

Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate of 0.06 rads. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is .

How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft from the wall?

The top of a ladder slides down a vertical wall at a rate of . At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of . How long is the ladder?

A faucet is filling a hemispherical basin of diameter 60 cm with water at a rate of . Find the rate at which the water is rising in the basin when it is half full. [Use the following facts: 1 L is . The volume of the portion of a sphere with radius from the bottom to a height is , as we will show in Chapter 6.]

. Boyles Law states that when a sample of gas is compressed at a constant temperature, the pressure and volume satisfy the equation , where is a constant. Suppose that at a certain instant the volume is 600 cm , the pressure is 150 kPa, and the pressure is increasing at a rate of 20 kPamin. At what rate is the volume decreasing at this instant?

When air expands adiabatically (without gaining or losing heat), its pressure and volume are related by the equation , where is a constant. Suppose that at a certain instant the volume is 400 cm and the pressure is 80 kPa and is decreasing at a rate of 10 kPamin. At what rate is the volume increasing at this instant?

If two resistors with resistances and are connected in parallel, as in the figure, then the total resistance , measured in ohms ( ), is given by If and are increasing at rates of and ,respectively, how fast is changing when and

Brain weight as a function of body weight in fish has been modeled by the power function , where and are measured in grams. A model for body weight as a function of body length (measured in centimeters) is . If, over 10 million years, the average length of a certain species of fish evolved from 15 cm to 20 cm at a constant rate, how fast was this species brain growing when the average length was 18 cm?

Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of . How fast is the length of the third side increasing when the angle between the sides of fixed length is 60 ?

Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley (see the figure). The point is on the floor 12 ft directly beneath and between the carts. Cart A is being pulled away from at a speed of 2 fts. How fast is cart B moving toward at the instant when cart A is 5 ft from ?

A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Lets assume the rocket rises vertically and its speed is 600 fts when it has risen 3000 ft. (a) How fast is the distance from the television camera to the rocket changing at that moment? (b) If the television camera is always kept aimed at the rocket, how fast is the cameras angle of elevation changing at that same moment?

A lighthouse is located on a small island 3 km away from the nearest point on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from ?

A plane flies horizontally at an altitude of and passes directly over a tracking telescope on the ground. When the angle of elevation is , this angle is decreasing at a rate of . How fast is the plane traveling at that time?

A Ferris wheel with a radius of is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level?

A plane flying with a constant speed of 300 kmh passes over a ground radar station at an altitude of 1 km and climbs at an angle of 30 . At what rate is the distance from the plane to the radar station increasing a minute later?

Two people start from the same point. One walks east at 3 mih and the other walks northeast at 2 mih. How fast is the distance between the people changing after 15 minutes?

A runner sprints around a circular track of radius 100 m at a constant speed of 7 ms. The runners friend is standing at a distance 200 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 m?

The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one oclock?