Question Video: Finding the Rate of Change of the Radius of an Expanding Cylinder given the Rate of Change of Its Surface Area Using Related Rates | Nagwa Question Video: Finding the Rate of Change of the Radius of an Expanding Cylinder given the Rate of Change of Its Surface Area Using Related Rates | Nagwa

# Question Video: Finding the Rate of Change of the Radius of an Expanding Cylinder given the Rate of Change of Its Surface Area Using Related Rates Mathematics • Third Year of Secondary School

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The height of a cylinder is equal to its base diameter. Maintaining this relationship between height and base diameter, the cylinder expands such that the rate of increase of its surface area is 32π cmΒ²/s with respect to time. Calculate the rate of increase of its radius when its base has a radius of 18 cm.

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### Video Transcript

The height of a cylinder is equal to its base diameter. 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. Calculate the rate of increase of its radius when its base has a radius of 18 centimetres.

The first thing we should do when dealing with related rates questions is identify what weβve been given and what weβre looking to find. In this question, weβve been given a cylinder whose height is equal to its base diameter. 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. Weβre also told that the rate of increase of its surface area is 32π square centimetres per second with respect to time. We know that the rate of change of something is considered to be its derivative.

So, letting π  be equal to the surface area, then we know that dπ  by dπ‘ is equal to 32π. We want to find the rate of increase of its radius. So, weβre looking to find dπ by dπ‘. So, how do we link dπ  by dπ‘ with dπ by dπ‘? Well, weβll use the chain rule. We can see that dπ  by dπ‘ will be equal to dπ  by dπ times dπ by dπ‘. 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. We know dπ  by dπ‘, but how are we going to work out dπ  by dπ?

Well, we recall the formula for the surface area of a cylinder. Itβs the area of the two circles, thatβs two ππ squared, plus the area of the rectangle in between, and thatβs two ππ times β. We already said, though, that the height of our cylinder is two π, so letβs replace β with two π in our formula. Simplifying, we find that the surface area of our cylinder is six ππ squared. And we see that we can find an expression for dπ  by dπ by differentiating this expression with respect to π. The first derivative of six ππ squared is two times six ππ, which is 12ππ.

Weβre looking to find the rate of increase when the radius is 18 centimetres, so weβll evaluate dπ  by dπ when π is equal to 18. Itβs 12π times 18, which is 216π. So, we now have dπ  by dπ‘ and dπ  by dπ. We said the dπ by dπ‘ was the quotient of these; itβs 32π over 216π. Which simplifies to four twenty-sevenths. 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.

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