Key Concepts and Formulas

• Surface Area of a Sphere: The surface area $S$ of a sphere with radius $r$ is given by $S = 4\pi r^2$.
• Related Rates: This involves finding the relationship between the rates of change of two or more variables that are related to each other.
• Chain Rule: $\frac{d}{dt}f(g(t)) = f'(g(t)) \cdot g'(t)$.
• Integration: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where C is the constant of integration.

Step-by-Step Solution

Step 1: Expressing the Rate of Change of Surface Area

We are given that the surface area of the spherical balloon increases at a constant rate. We need to express the rate of change of the surface area with respect to time.

• Formula for Surface Area: $S = 4\pi r^2$
• Differentiating with respect to time ($t$): Since both $S$ and $r$ are functions of time, we differentiate both sides of the equation with respect to $t$ using the chain rule. $\frac{dS}{dt} = \frac{d}{dt}(4\pi r^2)$ $\frac{dS}{dt} = 4\pi \frac{d}{dt}(r^2)$ $\frac{dS}{dt} = 4\pi (2r) \frac{dr}{dt}$ $\frac{dS}{dt} = 8\pi r \frac{dr}{dt}$ Here, $\frac{dS}{dt}$ is the rate of change of the surface area, and $\frac{dr}{dt}$ is the rate of change of the radius.

Step 2: Incorporating the Constant Rate Condition and Separating Variables

The problem states that the surface area increases at a constant rate. Let's denote this constant rate by $k$.

• Setting the rate to a constant: $\frac{dS}{dt} = k$ where $k$ is a constant.
• Substituting into the derived equation: $k = 8\pi r \frac{dr}{dt}$
• Separating variables: To solve the differential equation, we separate the variables $r$ and $t$. $k dt = 8\pi r dr$

Step 3: Integrating to Find the Relationship Between Radius and Time

Now that the variables are separated, we integrate both sides of the equation.

• Integrating both sides: $\int k dt = \int 8\pi r dr$
• Performing the integration: $kt = 8\pi \frac{r^2}{2} + C$ $kt = 4\pi r^2 + C$ Here, $C$ is the constant of integration.

Step 4: Using Initial Conditions to Determine Constants $C$ and $k$

We are given two conditions that will allow us to find the values of $C$ and $k$.

• Condition 1: At $t=0$ seconds, the radius $r=3$ units. Substitute $t=0$ and $r=3$ into the equation $kt = 4\pi r^2 + C$: $k(0) = 4\pi (3^2) + C$ $0 = 36\pi + C$ $C = -36\pi$ Now, our equation becomes: $kt = 4\pi r^2 - 36\pi \quad \text{... (Equation 1)}$

• Condition 2: After 5 seconds ($t=5$), the radius $r=7$ units. Substitute $t=5$ and $r=7$ into Equation 1: $k(5) = 4\pi (7^2) - 36\pi$ $5k = 4\pi (49) - 36\pi$ $5k = 196\pi - 36\pi$ $5k = 160\pi$ $k = \frac{160\pi}{5}$ $k = 32\pi$

• Finalized Relationship between $r$ and $t$: Now that we have both $C$ and $k$, substitute their values back into Equation 1: $32\pi t = 4\pi r^2 - 36\pi$

  Dividing the entire equation by $4\pi$ to simplify: $8t = r^2 - 9$

Step 5: Calculating the Radius After 9 Seconds

We need to find $r$ when $t=9$.

• Substitute $t=9$ into the simplified equation: $8(9) = r^2 - 9$ $72 = r^2 - 9$
• Solve for $r^2$: $r^2 = 72 + 9$ $r^2 = 81$
• Solve for $r$: $r = \sqrt{81}$ $r = 9$ Since radius must be a positive quantity, we take the positive root.

Thus, the radius of the balloon after 9 seconds is 9 units.

Common Mistakes & Tips

• Forgetting the Chain Rule: When differentiating $4\pi r^2$ with respect to $t$, remember to multiply by $\frac{dr}{dt}$.
• Omitting the Constant of Integration ($C$): This is crucial for correctly applying initial conditions.
• Simplifying the equation: Look for opportunities to simplify the equation (e.g., dividing by a common factor) to reduce calculation errors.

Summary

We found the relationship between the rate of change of the surface area and the radius of the balloon. By using the given constant rate and initial conditions, we derived an equation relating the radius and time. Finally, we used this equation to calculate the radius of the balloon after 9 seconds.

The final answer is $\boxed{9}$, which corresponds to option (A).