Q. 15 - Calculus

Question

Suppose the radius r, height h, and volume V of a cylinder are functions of time t. How is $\frac{dV}{dt}$ related to $\frac{dr}{dt}$ if the height of the cylinder is constant? 

Step-by-Step Solution

Answer

The derivatives $\frac{dV}{dt}$ and $\frac{dr}{dt}$ are related by $\frac{dV}{dt}=2\mathrm{\pi rh}\frac{\mathrm{dr}}{\mathrm{dt}}$.

Step 1. Formula used.

The volume of the sphere is given by the formula $V={\mathrm{\pi r}}^2\mathrm{h}$ cu. units.

Step 2. Apply the differentiation.

Given that the radius r, height h, and volume V are functions of t.

Apply the differentiation to $V={\mathrm{\pi r}}^2\mathrm{h}$ with respect to t  and keeping h as constant as follows.

$\frac{dV}{dt}=2\mathrm{\pi rh}\frac{\mathrm{dr}}{\mathrm{dt}}$

Step 3. Conclusion.

The derivatives $\frac{dV}{dt}$ and $\frac{dr}{dt}$ are related by $\frac{dV}{dt}=2\mathrm{\pi rh}\frac{\mathrm{dr}}{\mathrm{dt}}$.