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Air is being pumped into a spherical balloon at a rate of $100 \frac{cm^3}{s}$. Determine the rate at which the radius is increasing when the radius of the balloon is 50 cm.
Air is being pumped into a spherical balloon at a rate of $100 \frac{cm^3}{s}$. Determine the rate at which the radius is increasing when the radius of the balloon is 50 cm.
The radius is increasing at a rate of $\frac{1}{25\pi}$ cm/s.
The length of a rectangle increases at a rate of $8 \frac{cm}{sec}$, and its width increases at a rate of $3 \frac{cm}{sec}$. When the length is 20cm and the width is 10cm, at what rate is the area of the rectangle increasing?
The length of a rectangle increases at a rate of $8 \frac{cm}{sec}$, and its width increases at a rate of $3 \frac{cm}{sec}$. When the length is 20cm and the width is 10cm, at what rate is the area of the rectangle increasing?
The area of the rectangle is increasing at a rate of 140 $\frac{cm^2}{sec}$.
Two cars start at the same point. One travels south at $60 \frac{mi}{hr}$ and the other car travels west at $25 \frac{mi}{hr}$. After 2 hours, what is the rate at which the distance between the cars is increasing?
Two cars start at the same point. One travels south at $60 \frac{mi}{hr}$ and the other car travels west at $25 \frac{mi}{hr}$. After 2 hours, what is the rate at which the distance between the cars is increasing?
The distance between the cars is increasing at a rate of $65 \frac{mi}{hr}$.
A 10ft ladder rests against a wall. The bottom of the ladder slides away from the wall at a rate of $4 \frac{ft}{s}$. Determine the rate at which the top of the ladder is sliding down the wall when the bottom is 6 ft from the wall.
A 10ft ladder rests against a wall. The bottom of the ladder slides away from the wall at a rate of $4 \frac{ft}{s}$. Determine the rate at which the top of the ladder is sliding down the wall when the bottom is 6 ft from the wall.
A cylindrical tank with radius 5 meters is being filled with water at a rate of 3 cubic meters per minute. How fast is the height of the water increasing?
A cylindrical tank with radius 5 meters is being filled with water at a rate of 3 cubic meters per minute. How fast is the height of the water increasing?
A conical tank is 10 ft across the top and 12 ft deep. If water is flowing into the tank at a rate of 10 cubic ft/min, find the rate of change of the depth of the water when the water is 8 ft deep.
A conical tank is 10 ft across the top and 12 ft deep. If water is flowing into the tank at a rate of 10 cubic ft/min, find the rate of change of the depth of the water when the water is 8 ft deep.
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 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?
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 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?
Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute. It forms a conical pile whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 feet high?
Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute. It forms a conical pile whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 feet high?
A man walks along a straight path at a speed of 4 ft/s. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight?
A man walks along a straight path at a speed of 4 ft/s. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight?
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?
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?
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 1 meter per second, how fast is the area of the spill increasing when the radius is 30 meters?
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 1 meter per second, how fast is the area of the spill increasing when the radius is 30 meters?
A particle moves along the curve $y = \sqrt{x}$. As the particle passes through the point (4, 2), its x-coordinate increases at a rate of 3 cm/s. How fast is the distance from the particle to the origin changing at this instant?
A particle moves along the curve $y = \sqrt{x}$. As the particle passes through the point (4, 2), its x-coordinate increases at a rate of 3 cm/s. How fast is the distance from the particle to the origin changing at this instant?
A streetlight is mounted on a pole 20 feet high. A man 6 feet tall walks away from the pole at a rate of 5 feet per second. How fast is the tip of his shadow moving away from the pole when the man is 40 feet from the pole?
A streetlight is mounted on a pole 20 feet high. A man 6 feet tall walks away from the pole at a rate of 5 feet per second. How fast is the tip of his shadow moving away from the pole when the man is 40 feet from the pole?
Water is leaking out of an inverted conical tank at a rate of 10000 $\text{cm}^3$/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 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank.
Water is leaking out of an inverted conical tank at a rate of 10000 $\text{cm}^3$/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 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank.
A hot air balloon is rising vertically from a point on the ground 30 meters from an observer. If the balloon is rising at a rate of 3 meters per second, how fast is the angle of elevation from the observer to the balloon changing when the balloon is 40 meters high?
A hot air balloon is rising vertically from a point on the ground 30 meters from an observer. If the balloon is rising at a rate of 3 meters per second, how fast is the angle of elevation from the observer to the balloon changing when the balloon is 40 meters high?
A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 $\text{ft}^3$/min, how fast is the water level rising when the water has a depth of 8 inches?
A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 $\text{ft}^3$/min, how fast is the water level rising when the water has a depth of 8 inches?
A point P is moving along the line $y = 2x$. How fast is the distance between P and the point (3, 0) changing when P is at (4, 8) and its x-coordinate is increasing at a rate of 1 cm/sec?
A point P is moving along the line $y = 2x$. How fast is the distance between P and the point (3, 0) changing when P is at (4, 8) and its x-coordinate is increasing at a rate of 1 cm/sec?
A plane flying horizontally at an altitude of 2 miles and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 3 miles away from the station.
A plane flying horizontally at an altitude of 2 miles and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 3 miles away from the station.
Consider a rectangle with one side on the x-axis and two vertices on the curve $y = e^{-x^2}$. Find how fast the area of the rectangle is changing when $x = 1$ and x is increasing at a rate of 0.2.
Consider a rectangle with one side on the x-axis and two vertices on the curve $y = e^{-x^2}$. Find how fast the area of the rectangle is changing when $x = 1$ and x is increasing at a rate of 0.2.
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 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?
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 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?
Flashcards
Balloon Radius Increase
Balloon Radius Increase
The rate at which the radius of a spherical balloon is increasing, given the volume increases at 100 cm³/s, when the radius is 50 cm.
Rectangle Area Increase
Rectangle Area Increase
The rate at which the area of a rectangle is increasing, given the rate of increase of its length (8 cm/sec) and width (3 cm/sec) when the length is 20 cm and width is 10 cm.
Cars Distance Increase
Cars Distance Increase
The rate at which the distance between two cars is increasing 2 hours after they start moving from the same point; one travels south at 60 mi/hr, the other west at 25 mi/hr.
Ladder Sliding Down Wall
Ladder Sliding Down Wall
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Study Notes
- These problems all involve related rates, where the rate of change of one quantity is related to the rate of change of another.
Spherical Balloon Problem
- Air is being pumped into a spherical balloon at a rate of 100 cm³/s
- The rate at which the radius of the balloon is increasing is to be determined
- The radius of the balloon is 50 cm
Rectangle Area Problem
- The length of a rectangle is increasing at a rate of 8 cm/s
- The width of the rectangle is increasing at a rate of 3 cm/s
- The length is 20 cm
- The width is 10 cm
- Find how fast the area of the rectangle is increasing
Two Cars Problem
- Two cars start from the same point.
- One car travels south at 60 mi/hr.
- The other car travels west at 25 mi/hr.
- Determine the rate at which the distance between the cars is increasing 2 hours later.
Ladder Problem
- A 10ft ladder rests against a vertical wall.
- The bottom of the ladder slides away from the wall at a rate of 4 ft/s.
- Determine how fast the top of the ladder is sliding down the wall.
- The bottom of the ladder is 6 ft from the wall.
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