12/11 Related Rates

A related rate problem involves finding the rate at which one quantity changes with respect to another quantity that is also changing. These problems often involve the use of derivatives and the chain rule.

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

To find the rate of change of volume with respect to radius, differentiate the volume formula with respect to time: $$\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt}$$ .

The derivative of the volume V of a sphere with respect to its radius r is: $$\frac{dV}{dr} = 4\text{π}r^2$$ .

This can be expressed as: $$\frac{dr}{dt} = 2 \text{ in/sec}$$ .

Using the chain rule: $$\frac{dV}{dt} = 4\text{π}(2)^2 \times 2 = 32\text{π} \text{ in}^3/\text{sec}$$ .

The area A of a circle with radius r is given by the formula: $$A = \text{π}r^2$$ .

To find the rate of change of area with respect to radius, differentiate the area formula with respect to time: $$\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}$$ .

The derivative of the area A of a circle with respect to its radius r is: $$\frac{dA}{dr} = 2\text{π}r$$ .

This can be expressed as: $$\frac{dr}{dt} = 3 \text{ cm/min}$$ .