Question

Surface Area The radius \(r\) and surface area \(S\) of a sphere are related by the equation \(S=4 \pi r^{2} .\) Write an equation that relates \(d S / d t\) to $d r / d t .

Short answer

The equation that relates \(d S / d t\) to \(d r / d t\) is \(d S / d t = 8 \pi r * d r / d t\).

Step-by-step solution

Identify the original equation

The original equation given is: \(S=4 \pi r^{2} .\)

Apply the power rule of differentiation

Differentiating both sides of the equation with respect to \(t\), we get: \(d S / d t = d(4 \pi r^{2}) / d t\). Applying the power rule and chain rule of differentiation (because \(r\) is a function of \(t\)), the right side of the equation becomes: \(d S / d t = 4 \pi * 2r * d r / d t\).

Simplify the equation

Simplify the equation to make it clearer by multiplying out the constants, giving: \(d S / d t = 8 \pi r * d r / d t\).