a. find dr/dh for a spherical cap with a volume of 2π. what is the best way to find dr/dh? a. use implicit…

Explanation:

Step1: Recall implicit - differentiation rule

When differentiating an equation with respect to a variable, we treat other variables as functions of that variable. For the volume formula of a spherical cap $V=\frac{\pi h^{2}(3r - h)}{3}=\pi r h^{2}-\frac{\pi h^{3}}{3}$, if we want to find $\frac{dr}{dh}$, we differentiate both sides of the equation with respect to $h$.

Step2: Differentiate the left - hand side

Since $V$ is a constant (in the context of finding the relationship between $r$ and $h$ for a given volume), $\frac{dV}{dh} = 0$.

Step3: Differentiate the right - hand side

Using the product rule and power rule. The derivative of $\pi r h^{2}$ with respect to $h$ is $\pi\left(2h r+h^{2}\frac{dr}{dh}\right)$ and the derivative of $-\frac{\pi h^{3}}{3}$ with respect to $h$ is $-\pi h^{2}$. So, $\frac{d}{dh}\left(\pi r h^{2}-\frac{\pi h^{3}}{3}\right)=\pi\left(2h r + h^{2}\frac{dr}{dh}\right)-\pi h^{2}$.

Step4: Set up the equation

Set $\frac{dV}{dh}$ equal to the derivative of the right - hand side: $0=\pi\left(2h r+h^{2}\frac{dr}{dh}\right)-\pi h^{2}$.

Step5: Solve for $\frac{dr}{dh}$

First, expand the equation: $0 = 2\pi h r+\pi h^{2}\frac{dr}{dh}-\pi h^{2}$. Then, isolate $\frac{dr}{dh}$: [ \begin{align*} \pi h^{2}\frac{dr}{dh}&=\pi h^{2}- 2\pi h r\ \frac{dr}{dh}&=\frac{h - 2r}{h}=1-\frac{2r}{h} \end{align*} ]

The best way to find $\frac{dr}{dh}$ is to use implicit differentiation to find $\frac{dr}{dh}$. Differentiate both sides of the equation with respect to $h$, treating $r$ as a function of $h$. Then solve for $\frac{dr}{dh}$.

Answer:

A. Use implicit differentiation to find $\frac{dr}{dh}$. Differentiate both sides of the equation with respect to $h$, treating $r$ as a function of $h$. Then solve for $\frac{dr}{dh}$.