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The height of a cylinder is equal to its base diameter.
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Maintaining this relationship between height and base diameter, the cylinder expands such that the rate of increase of its surface area is 32π square centimetres per second with respect to time.
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Calculate the rate of increase of its radius when its base has a radius of 18 centimetres.
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The first thing we should do when dealing with related rates questions is identify what weβve been given and what weβre looking to find.
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In this question, weβve been given a cylinder whose height is equal to its base diameter.
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Letting π be equal to the radius of the cross section of this cylinder, we find its base diameter and its height is equal to two π.
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Weβre also told that the rate of increase of its surface area is 32π square centimetres per second with respect to time.
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We know that the rate of change of something is considered to be its derivative.
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So, letting π be equal to the surface area, then we know that dπ by dπ‘ is equal to 32π.
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We want to find the rate of increase of its radius.
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So, weβre looking to find dπ by dπ‘.
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So, how do we link dπ by dπ‘ with dπ by dπ‘?
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Well, weβll use the chain rule.
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We can see that dπ by dπ‘ will be equal to dπ by dπ times dπ by dπ‘.
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By dividing through by dπ by dπ, we find that dπ by dπ‘ will be calculated by dividing dπ by dπ‘ by dπ by dπ.
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We know dπ by dπ‘, but how are we going to work out dπ by dπ?
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Well, we recall the formula for the surface area of a cylinder.
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Itβs the area of the two circles, thatβs two ππ squared, plus the area of the rectangle in between, and thatβs two ππ times β.
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We already said, though, that the height of our cylinder is two π, so letβs replace β with two π in our formula.
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Simplifying, we find that the surface area of our cylinder is six ππ squared.
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And we see that we can find an expression for dπ by dπ by differentiating this expression with respect to π.
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The first derivative of six ππ squared is two times six ππ, which is 12ππ.
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Weβre looking to find the rate of increase when the radius is 18 centimetres, so weβll evaluate dπ by dπ when π is equal to 18.
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Itβs 12π times 18, which is 216π.
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So, we now have dπ by dπ‘ and dπ by dπ.
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We said the dπ by dπ‘ was the quotient of these; itβs 32π over 216π.
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Which simplifies to four twenty-sevenths.
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The rate of increase of the radius of our cylinder when its base has a radius of 18 centimetres is four twenty-sevenths centimetres per second.