 Hi and welcome to the session. Let us discuss the following question. Question says the volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units, find the radius of balloon after 3 seconds. Let us now start with the solution. First of all let us assume that the volume of the spherical balloon is v cubic units. So here we can write let the volume of the spherical balloon is equal to v cubic units. Now we are given that volume of the spherical balloon changes at a constant rate. Now if the volume of the spherical balloon is v cubic units, then rate of change of volume is equal to k where k is any constant. Now this further implies dv is equal to k dt. Now integrating both the sides of this equation we get integral of dv is equal to k multiplied by integral of dt. Now using this formula of integration we get integral of dv is equal to v and integral of dt is equal to t. So here we can write kt plus c where c is the constant of integration. Let us now name this equation as 1. Now let us assume that r be the radius of the spherical balloon. Then volume v of spherical balloon is equal to 4 upon 3 pi r cube. Now substituting value of v is equal to 4 upon 3 pi r cube. In equation 1 we get 4 upon 3 pi r cube is equal to kt plus c. Now we are given that initially the radius of the balloon is 3 units. So we can say when t is equal to 0 by t we mean time. When time is equal to 0 r is equal to 3. Now let us name this equation as 2. Now putting t is equal to 0 and r is equal to 3. In equation 2 we get 4 upon 3 pi multiplied by 3 cube is equal to k multiplied by 0 plus c. Now this further implies 36 pi is equal to c. We are also given that after 3 seconds radius is 6 units. So we can say when t is equal to 3 r is equal to 6. Now putting t is equal to 3 and r is equal to 6. In equation 2 we get 4 upon 3 pi multiplied by 6 cube is equal to k multiplied by 3 plus c. Now from this equation we know c is equal to 36 pi. Now we will substitute this value of c in this equation. We get 4 upon 3 pi multiplied by 6 cube is equal to 3k plus 36 pi. Now simplifying this term we get 288 pi 288 pi is equal to 3k plus 36 pi. Now subtracting 36 pi from both the sides of this equation we get 288 pi minus 36 pi is equal to 3k. Now this further implies 252 pi is equal to 3k. Subtracting these two terms we get 252 pi. Now dividing both the sides pi 3 we get 252 pi upon 3 is equal to k. Now this further implies 84 pi is equal to k. Or we can simply write it as k is equal to 84 pi. Now we know this is the equation 2. Now substituting corresponding values of k and c in this equation we get 4 upon 3 pi r cube is equal to 84 pi multiplied by t plus 36 pi. Now dividing both the sides of this equation by pi we get 4 upon 3 r cube is equal to 84 t plus 36. Now dividing both the sides of this equation by 4 we get r cube upon 3 is equal to 21 t plus 9. Now multiplying both the sides of this equation by 3 we get r cube is equal to 63 t plus 27. Now this further implies r is equal to cube root of 63 t plus 27. Or we can say r is equal to 63 t plus 27 raise to power 1 upon 3. So this is the required radius. So we get the required radius of the balloon after these seconds is 63 t plus 27 raise to the power 1 upon 3 units. This is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.