The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If in

Q: Wait, why do we integrate the rate of change of surface area? Can't we just differentiate the radius?

That is a common trap! The problem explicitly gives us the rate of change of the *surface area*, not the radius. If we were given the rate of change of the radius, we would differentiate. Since we are given the rate of change of the area, we must integrate the constant rate to find the area function $S(t)$, and then relate it back to the radius using $S = 4\pi r^2$.

Q: Why is the constant of integration $C$ not zero?

The constant of integration $C$ represents the initial state of the system at $t=0$. In this problem, the balloon already has a radius of 3 units at $t=0$, so the surface area is not zero. Therefore, $C$ must be the initial surface area, $4\pi(3)^2 = 36\pi$.

Q: Is it possible for the radius to be negative?

Mathematically, when we solve $r^2 = 81$, we get $r = \pm 9$. However, in the physical world, the radius of a balloon must be a positive quantity. Thus, we reject the negative root and accept only $r = 9$.

Question

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 :

Options:

(A) 9

(B) 10

(C) 11

(D) 12

Sigma Percentile · Accepted Answer

### The Geometry of Growth: A Calculus Journey

Imagine you are holding a spherical balloon. As you pump air into it, the balloon expands. It is a simple, everyday sight, but beneath this expansion lies the elegant language of calculus.

Today, we are going to peel back the layers of this process to understand how the radius of a balloon changes when its surface area grows at a constant rate.

### Phase 1: The Geometry of the Sphere

First, let us ground ourselves in the geometry. For any sphere, the surface area $S$ is intrinsically linked to its radius $r$ by the formula:

$$S = 4\pi r^2$$

This is our bridge. Whenever we talk about the surface area, we are implicitly talking about the radius. Keep this relationship in your mind; it is the key that will unlock the entire problem.

### Phase 2: The Calculus of Change

The problem gives us a fascinating piece of information: the surface area increases at a constant rate. In the language of calculus, this means the derivative of the surface area with respect to time is a constant, $k$:

$$\frac{dS}{dt} = k$$

This is our governing differential equation. It tells us that for every second that passes, the surface area adds a fixed amount of space to its total.

To find the total surface area at any time $t$, we simply integrate both sides with respect to time:

$$\int dS = \int k dt$$

This gives us the linear equation for surface area:

$$S(t) = kt + C$$

Here, $C$ is the constant of integration, representing the initial surface area of the balloon at $t=0$.

### Phase 3: Solving for the Constants

Now, we combine our geometric bridge with our calculus result. Substituting $S = 4\pi r^2$ into our equation, we get:

$$4\pi r^2 = kt + C$$

We have two unknowns: $k$ and $C$. We use the data provided to find them. At $t=0$, the radius $r=3$. Plugging these into our equation:

$$4\pi (3)^2 = k(0) + C \Rightarrow C = 36\pi$$

With $C$ found, we turn to the second clue: at $t=5$, the radius $r=7$. Substituting these values:

$$4\pi (7)^2 = k(5) + 36\pi$$

$$196\pi = 5k + 36\pi$$

$$160\pi = 5k \Rightarrow k = 32\pi$$

### Phase 4: The Final Calculation

We have successfully decoded the growth of this balloon. Our general equation for the radius at any time $t$ is:

$$4\pi r^2 = 32\pi t + 36\pi$$

Dividing by $4\pi$ simplifies this beautifully to:

$$r^2 = 8t + 9$$

Now, the moment of truth. We want to find the radius at $t=9$ seconds. We simply substitute $t=9$ into our simplified equation:

$$r^2 = 8(9) + 9$$

$$r^2 = 72 + 9 = 81$$

Taking the square root, we find **$r=9$**. The balloon, having started with a radius of 3, has grown to a radius of 9 units in 9 seconds. It is a perfect, clean result—a testament to the power of calculus to describe the world around us.

Animated Solution for Mathematics - Differentiation: 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 :

Select Answer:

Visualized Solution

Visualizing the Inflation

Defining the Rate of Change

Integrating the Rate Equation

Substituting Surface Area Formula

Applying Initial Conditions ( $t = 0$ )

Using the Second Condition ( $t = 5$ )

Solving for the Rate Constant $k$

The General Radius Equation

Finding Radius at $t = 9$

Final Calculation

The Sigma Insight: Tangents, Normals and Rate Measure

The Geometry of Growth

Phase 1

Phase 2

Phase 3

Phase 4

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