Related Rates

Related Rates are a method used in calculus to find the rate at which one quantity changes in relation to another quantity that is also changing.

The volume of a sphere is given by the formula: $$V = \frac{4}{3}\pi r^3$$ where V is the volume and r is the radius.

Use the formula for the area of a circle, $$A = \pi r^2$$ . Differentiate both sides with respect to time to relate $$\frac{dA}{dt}$$ and $$\frac{dr}{dt}$$ .

As the radius increases, the volume increases at a rate proportional to the square of the radius, due to the volume formula $$V = \frac{4}{3}\pi r^3$$ .

Differentiate the volume formula $$V = \frac{4}{3}\pi r^3$$ with respect to time to find $$\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$$ .

Negative rates indicate a decrease in the quantity being measured, such as a shrinking radius or decreasing volume.

Use the Pythagorean theorem to relate the distances between moving objects, then differentiate with respect to time to find rates of change.

The volume of a cone is given by the formula: $$V = \frac{1}{3} \pi r^2 h$$ where r is the radius and h is the height.

Use the volume formula for a cone, differentiate with respect to time, and relate the rates of change of volume and height.

The rate of change of the distance between two objects is the sum of their individual rates of change, considering their directions.