 Hello and welcome to the session. Let us understand the following question which says, the volume of a cube is increasing at the rate of 8 cm cube per second. How fast is the surface area increasing when the length of an edge is 12 cm? Now let us proceed on to the solution. Let X be the length of the cube. Then volume V is equal to X cube The surface area S is equal to 6 X square where X is the function of time t and is given to us that volume of a cube is increasing at the rate of 8 cm cube per second that is dV by dt is equal to 8 and we have to find how fast is the surface area increasing which means we have to find dS by dt dV by dt can be written as dV by dx multiplied by dx by dt by chain root. Let us name this equation as 1. It is given to us that V is equal to X cube and S is equal to 6 X square Now differentiating V with respect to X we get dV by dx is equal to 3 X square Now similarly differentiating S with respect to t we get dS by dt is equal to d by dt of 6 X square which is equal to 12 X d by dt of X therefore dS by dt is equal to 12 X multiplied by dx by dt Let us name this as equation 2 and this as equation 3 From equation 1 we have dV by dt is equal to dV by dx multiplied by dx by dt dV by dt is given to us as 8 So this implies 8 is equal to dV by dx is equal to 3 X square from 2 dx by dt is unknown which implies dx by dt is equal to 8 by 3 X square Now let us name this equation as 4 Now substituting dx by dt from 4 in 3 we get dS by dt is equal to 12 X multiplied by 8 by 3 X square Here this X gets cancelled with this X and 12 gets cancelled by c and we get here 4 So finally we get 4 aids are 32 divided by X and we have to find the surface area increasing When the length of the edge is 12 cm Therefore dS by dt at X is equal to 12 is equal to 32 divided by 12 This gets cancelled by 4 aids are 32 and 4 speeds are 12 So it is equal to 8 by 3 cm square per second Hence d required answer the surface area is increasing at the rate of 8 by 3 cm square per second when edge is 12 cm I hope you understood this question Bye and have a nice day