A cylindrical water tank is expanding due to the increasing water pressure. Both the radius of the base and the height of the cylinder are changing over time. The lateral surface area of the cylinder is given by the formula: S = 2 π r h S=2\pi rh If the radius and height of the tank vary with time, how can we express the rate of change of the lateral surface area ( d S d t \frac{dS}{dt} ) in terms of the rates of change of the radius ( d r d t \frac{dr}{dt} ) and the height ( d h d t \frac{dh}{dt} )?

d S d t = 2 π ( h ⋅ d r d t + r ⋅ d h d t ) \frac{dS}{dt}=2\pi\left(h\cdot\frac{dr}{dt}+r\cdot\frac{dh}{dt}\right)

d S d t = 2 π ( d r d t + d h d t ) \frac{dS}{dt}=2\pi\left(\frac{dr}{dt}+\frac{dh}{dt}\right)

d S d t = π ( h ⋅ d r d t + r ⋅ d h d t ) \frac{dS}{dt}=\pi\left(h\cdot\frac{dr}{dt}+r\cdot\frac{dh}{dt}\right)

d S d t = π ( d r d t + d h d t ) \frac{dS}{dt}=\pi\left(\frac{dr}{dt}+\frac{dh}{dt}\right)

A cylindrical water tank is expanding due to the increasing water...

0. Functions(0)

1. Limits and Continuity(0)

2. Intro to Derivatives(0)

3. Techniques of Differentiation(0)

4. Applications of Derivatives(0)

5. Graphical Applications of Derivatives(0)

6. Derivatives of Inverse, Exponential, & Logarithmic Functions(0)

7. Antiderivatives & Indefinite Integrals(0)

8. Definite Integrals(0)