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?

Short 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-by-step solution

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 oft.

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

role="math" localid="1648728564870" $\frac{dV}{dt}=2\mathrm{\pi rh}\frac{\mathrm{dr}}{\mathrm{dt}}$

Step 3. Conclusion.

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