The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :

Solution

Maths sir · 4 steps

Constant rate of surface area means r² grows linearly with time, so work with r², not r.

1. Set up the rate. Surface area is S = 4π r². A constant rate means dS/dt = c, so S, and therefore r², increases by equal amounts in equal times.

   4π r² = ct + 4π r₀² ⇒ r² = r₀² + kt

2. Find k. At t = 0, r² = 9. At t = 5, r² = 49. So 5k = 49 - 9 = 40, which gives k = 8 square units per second.
3. Go to 9 seconds. r² = 9 + 8 × 9 = 81, so r = 9 units.
4. So. The radius after 9 seconds is 9, option (a).

The idea to keep. Whatever quantity changes at a constant rate is the one that is linear in time. Here it is r², not r. Integrate the rate first, then fit the given data.

Answer from the NTA final answer key. Solution by PrepWiser.