Question

Volume The radius \(r,\) height \(h,\) and volume \(V\) of a right circular cylinder are related by the equation \(V=\pi r^{2} h .\) (a) How is \(d V / d t\) related to \(d h / d t\) if \(r\) is constant? (b) How is \(d V / d t\) related to \(d r / d t\) if \(h\) is constant? (c) How is \(d V / d t\) related to \(d r / d t\) and \(d h / d t\) if neither \(r\) nor \(h\) is constant?

Short answer

a) \(d V / d t = \pi r^{2} d h / d t\) \n b) \(d V / d t = 2 \pi r h d r / d t\) \n c) \(d V / d t = \pi r^{2} d h / d t + 2 \pi r h d r / d t\)

Step-by-step solution

(a) Derivative \(d V / d t\) when \( r \) is constant

Given that \( r \) is constant, we differentiate the volume equation with respect to \( t \). Differentiation in this case, with \( r \) constant, results to a constant multiplier to \( h \) hence \[d V / d t = \pi r^{2} d h / d t .\]

(b) Derivative \(d V / d t\) when \( h \) is constant

When \( h \) is constant, differentiating the volume equation with respect to \( t \) makes \( h \) a constant multiplier of \( r^2 \) in this case hence \[d V / d t = 2 \pi r h d r / d t .\]

(c) Derivative \(d V / d t\) when neither \( r \) nor \( h \) is constant

When neither \( r \) nor \( h \) is constant, we apply the product rule of differentiation to differentiate the volume equation with respect to \( t \). This results to \[d V / d t = \pi r^{2} d h / d t + 2 \pi r h d r / d t .\]