the volume V of a right circular cylinder of height 3 feet and radius r feet is V=V(r)=3(pi)(r^2). find the instantaneous rate o f change of the volume with respect to the radius at r=3.
1 answer:
When I see the words "instantaneous rate of change", I have to assume that you're in some stage of pre-calculus in your math class. The instantaneous rate of change of a function is just its first derivative. We have the function V(r) = 3 π r² and we need its first derivative with respect to ' r '. That shouldn't be too hard, because the ' 3 π ' is nothing but constants. Watch me while I do it slowly for you: -- The derivative of ' r² ' with respect to ' r ' is ' 2r '. -- The derivative of V(r) with respect to ' r ' is (3 π) times the derivative of ' r² '. -- The derivative of V(r) with respect to ' r ' is (3 π) times (2r) = <u>6 π r</u> . The value of the derivative when r=3 is (6 π 3) = 18π = about <em>56.5 feet³/foot .</em>
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