Related Rates - Conical Tank, Ladder Angle & Shadow Problem, Circle & Sphere - Calculus

This paragraph introduces the concept of related rates and derivatives in calculus. It begins with a discussion on derivatives, specifically focusing on the power rule for differentiation. The explanation includes examples of finding derivatives with respect to different variables, such as the derivative of y^3 with respect to x, and r^4 with respect to x. The paragraph emphasizes understanding the pattern of differentiation and the resulting derivatives. It then transitions into related rate problems, where the focus is on differentiating equations with respect to time. The first problem involves finding the rate of change of z given x, y, dx/dt, and dy/dt. The paragraph concludes with a practical example of finding the derivative of an equation involving x, y, and z, and how it relates to the Pythagorean theorem.

This paragraph delves into solving for rates of change in different scenarios using related rates. It starts with an example of finding the rate of change of the distance between the origin and a moving point on a graph. The paragraph explains the process of using implicit differentiation and plugging in given values to find the rate of change, denoted as dz/dt. The discussion then moves to a problem involving the radius of a circle decreasing at a certain rate, and how this affects the area and circumference of the circle. The paragraph also covers how to find the rate of change of the surface area of a snowball and the diameter of a sphere, highlighting the importance of understanding the units involved in the calculations.

This paragraph focuses on analyzing changes in geometric shapes using related rates. It begins with a problem involving the side length of a square increasing at a certain rate and how this affects the area and perimeter of the square. The paragraph then presents a problem about a balloon being inflated and how the volume change rate is related to the radius change rate. It also discusses the increase in surface area and volume of a cube with respect to the increase in its side length. The paragraph concludes with a problem involving a ladder leaning against a wall and how the rate of change of its position and the area of the triangle formed can be calculated.

This paragraph applies the concept of related rates to solve real-world problems. It starts with a scenario of a man walking away from a pole and how the tip of his shadow and the length of his shadow change. The paragraph then moves on to a problem involving a spotlight on the ground and a man walking towards a building, calculating how the length of his shadow on the building changes. The discussion includes setting up proportions and using similar triangles to find the rates of change. The paragraph also covers a problem involving two cars moving in different directions and how the rate of change of the distance between them can be calculated. The final problem involves two ships moving apart and how the rate of change of their distance is determined.

This paragraph discusses the calculation of rates of change in various motion scenarios. It begins with an example of an airplane traveling horizontally and how the distance between the plane and a radar station changes over time. The paragraph then describes how to find the rate at which the angle between the ground and an observer's line of sight to an airplane changes. It also covers a baseball player running towards first base and how the distance between him and second base changes. The paragraph concludes with a problem involving a trough filled with water and how the water level changes over time. The discussion includes using trigonometric functions, Pythagorean theorem, and differentiation to solve these problems.