Question Video: Finding an Expression That Represents the Rate of Change of a Cubeβs Volume Using Related Rates Mathematics • Higher Education

If π is the volume of a cube with edge length π₯ and the cube expands as time passes, give an expression for ππ/ππ‘.

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Video Transcript

If π is the volume of a cube with edge length π₯ and the cube expands as time passes, give an expression for ππ ππ‘.

So first of all, Iβm gonna take a look at what weβve actually given in the question. We know that π is the volume and π₯ is the edge length. So therefore, we can say that π is equal to π₯ cubed. So the volume is equal to π₯ cubed. And thatβs because if we tried to work out the volume of a cube, all we do is we actually cube one of the lengths of the sides. Okay, great, so weβve got our first little expression there.

So now, the next step would actually to be work out ππ ππ₯. So we actually gonna differentiate the value that we had earlier which was that π is equal to π₯ cubed. So ππ ππ₯ can be equal to three π₯ squared. And weβll reach that because if weβre multiplying the exponent by the coefficient, thatβs three multiplied by one which gives us three. And then, all weβve done is weβve actually reduced the exponent by one because the three minus one which gives us two. So we get three π₯ squared.

So now as weβre actually looking to find ππ ππ‘, weβre now gonna use the chain rule, which states that ππ¦ ππ₯ is equal to ππ¦ ππ‘ multiplied by ππ‘ ππ₯. Well, in our case, weβre gonna get ππ ππ‘ because this is like ππ¦ ππ₯, which is going to be equal to ππ ππ₯ multiplied by ππ₯ ππ‘. And we get that because if we take a look, our ππ₯ terms will actually cancel out because we will have a ππ₯ on the numerator, ππ₯ on the denominator. So then, weβll gonna get ππ ππ‘.

Okay, we already know ππ ππ₯ from our previous step. However, we donβt know ππ₯ ππ‘. So therefore, ππ ππ‘ what it actually means in practice, which is the change in the volume of the cube over time, is equal to three π₯ squared ππ₯ ππ‘.