Question

1. The volume of a cube is increasing at a rate of 1200 cm/min at the moment when the lengths of the sides are 20cm. How fast are the lengths of the sides increasing at that [10] moment?

Answer 1

`1\,\,cm/min`

Explanation:

Let V be the volume of cube and x be it's side .

We know that volume of cube is
`\left ( side \right )^(3)` i.e.,
`x^3`

Given :

`\frac{\mathrm{d} V}{\mathrm{d} t}=1200\,\,cm^3/min`

`x=20\,\,cm`

To find :
`\frac{\mathrm{d} x}{\mathrm{d} t}`

Solution :

Consider equation
`V=x^3`

On differentiating both sides with respect to t , we get

`\frac{\mathrm{d} V}{\mathrm{d} t}=3x^2\left ( \frac{\mathrm{d} x}{\mathrm{d} t} \right )\\1200=3(20)^2\left ( \frac{\mathrm{d} x}{\mathrm{d} t} \right )\\\frac{\mathrm{d} x}{\mathrm{d} t} =(1200)/(3(20)^2)=(1200)/(3* 400)=(1200)/(1200)=1\,\,cm/min`

So,

Length of the side is increasing at the rate of
`1\,\,cm/min`