12/11 Related Rates

A related rate problem involves finding the rate at which one quantity changes in relation to another quantity that is also changing. These problems often involve derivatives and require an understanding of how different variables are interconnected.

The area A of a circle is given by the formula: $$A = \frac{1}{2} \times \text{radius}^2$$ .

To find the rate of change of the area A with respect to the radius r, use the derivative: $$\frac{dA}{dr} = 2\text{π}r$$ .

If the radius r increases at a rate of dr/dt, then the area A changes at a rate of $$\frac{dA}{dt} = 2\text{π}r \frac{dr}{dt}$$ .

The volume V of a sphere is given by the formula: $$V = \frac{4}{3}\text{π}r^3$$ .

To find the rate of change of the volume V with respect to the radius r, use the derivative: $$\frac{dV}{dr} = 4\text{π}r^2$$ .

If the radius r increases at a rate of dr/dt, then the volume V changes at a rate of $$\frac{dV}{dt} = 4\text{π}r^2 \frac{dr}{dt}$$ .

The area of a circle increases with the square of the radius, meaning that as the radius increases, the area increases at a rate proportional to the square of the radius.

The volume of a sphere increases with the cube of the radius, meaning that as the radius increases, the volume increases at a rate proportional to the cube of the radius.

1. Identify the quantities that are changing and their relationships. 2. Write down the equations that relate these quantities. 3. Differentiate the equations with respect to time. 4. Substitute known values and solve for the unknown rate.