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

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

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: $$\frac{dr}{dt} = 2 \text{ in/sec}$$ .

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

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

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: $$\frac{dr}{dt} = 3 \text{ cm/min}$$ .

Substituting r = 2 in into the formula $$\frac{dV}{dt} = 4\text{π}r^2 \times \frac{dr}{dt}$$ gives $$\frac{dV}{dt} = 4\text{π}(2^2)(2) = 32\text{π} \text{ in}^3/\text{sec}$$ .