Ultimate Related Rates Study Guide (cars approaching an intersection, water into a tank, $ more)

This paragraph introduces a calculus problem involving a circular metal plate whose radius increases at a rate of 0.02 cm/min. The goal is to determine the rate at which the area of the plate is increasing when the radius is 60 cm. The explanation involves using the formula for the area of a circle (A = πr^2), taking the derivative with respect to time, and plugging in the given values to find the rate of change of the area (dA/dt). The problem is solved by applying the chain rule of differentiation and understanding the relationship between the radius and the area of a circle as functions of time.

The second paragraph discusses a related rates problem concerning a water tank shaped like an inverted circular cone. Given the base radius and height of the tank, and the rate at which water is pumped in, the task is to find the rate at which the water level is rising when the water depth is three meters. The solution involves understanding the volume formula for a cone (V = 1/3πr^2h), setting up an equation based on the given dimensions and rate of water flow, and differentiating this equation with respect to time to find the rate of change of the water level (dh/dt). The explanation also includes the use of similar triangles to relate the changing dimensions of the water-filled cone.

This paragraph continues the discussion on the related rates problem involving a cone-shaped water container. The focus is on differentiating the volume formula of the cone with respect to time and using the properties of similar triangles to simplify the equation. The explanation involves identifying the changing variables (radius and height of the water cone), setting up the volume equation, and applying the chain rule for differentiation. The solution requires algebraic manipulation to isolate the rate of change of the water level and understanding how the dimensions of the cone change as water is added.

The fourth paragraph presents a related rates problem involving a ladder leaning against a wall. The challenge is to determine how fast the top of the ladder is sliding down the wall when the bottom of the ladder moves away at a rate of one foot per second, and the bottom is two feet away from the wall. The solution involves creating a right triangle model, applying the Pythagorean theorem, and differentiating the equation with respect to time to find the rate of change of the vertical distance (dy/dt). The explanation emphasizes understanding the relationship between the changing horizontal and vertical distances and the constant length of the ladder.

This paragraph describes a related rates problem involving two cars moving towards an intersection. Car A is traveling west and Car B is traveling north, and the task is to find the rate at which the cars are approaching each other when Car A is 0.3 miles and Car B is 0.4 miles away from the intersection. The solution involves setting up a right triangle model based on the cars' positions and speeds, applying the Pythagorean theorem, and differentiating the equation with respect to time to find the rate of change of the distance between the cars (dC/dt). The explanation highlights the need to consider the direction of the cars' movement when determining the rate of change.

The sixth paragraph discusses a related rates problem involving two ships moving apart from each other. Ship A is moving west and Ship B is moving north, and the goal is to find the rate at which the distance between the ships is changing at 4 o'clock, given their initial positions and speeds. The explanation involves creating a right triangle model, applying the Pythagorean theorem, and differentiating the equation with respect to time to find the rate of change of the distance (dC/dt). The solution requires calculating the distances traveled by each ship and understanding how these distances change over time.

The ninth paragraph presents a related rates problem involving a triangle with two sides of fixed length and an angle between them that is increasing at a rate of 0.06 radians per second. The task is to find the rate at which the area of the triangle is increasing when the angle is π/3 radians. The solution involves using the formula for the area of a triangle (A = 1/2 * sides * sin(angle)), differentiating this formula with respect to time, and plugging in the given values to find the rate of change of the area (dA/dt). The explanation emphasizes understanding how the area of a triangle changes as the angle between its sides increases.

The eleventh paragraph describes a related rates problem involving an observer on the ground watching a balloon ascending vertically. The balloon is initially 500 feet away from the observer, and the task is to find the rate at which the angle of inclination of the observer's line of sight is increasing at the instant when the balloon is 400 feet above the observer's eyes. The solution involves setting up a right triangle with the observer's line of sight, applying the tangent function, and differentiating the equation with respect to time to find the rate of change of the angle (dθ/dt). The explanation highlights the relationship between the changing height of the balloon and the angle of inclination as functions of time.