The total surface area π of a right circular cylinder is related to the base radius r and height h by the equation π = 2ππ 2 + 2ππβ a) How is dS/dt related to dr/dt and dh/dt if neither r nor h is a constant? b) How is dS/dt related to dr/dt if h is constant? c) How is dS/dt related to dr/dt if r is constant? Solution: a) ππ π 2ππ 2 ππ π 2ππ ππ π 2ππ πβ = β + β ββ+ β βπ = ππ‘ ππ ππ‘ ππ ππ‘ πβ ππ‘ = 4ππ β ππ ππ πβ ππ πβ + 2πβ β + 2ππ β = 2π(2π + β) β + 2ππ β ππ‘ ππ‘ ππ‘ ππ‘ ππ‘ So ππ ππ πβ = 2π(2π + β) β + 2ππ β ππ‘ ππ‘ ππ‘ b) If h is constant then πβ = 0. ππ‘ Thus ππ ππ = 2π(2π + β) β ππ‘ ππ‘ c) If r is constant then ππ = 0. ππ‘ Thus ππ πβ = 2ππ β ππ‘ ππ‘
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