The problem describes a balloon in the shape of a right circular cylinder with hemispherical ends, with an initial radius of $r = 1.5 \, \text{m}$ and a length (excluding the hemispheres) of $h = 4 \, \text{m}$. The radius and length increase by $0.01 \, \text{m}$ and $0.05 \, \text{m}$, respectively. The task is to find the percentage change in the balloon's volume after these dimensional increases.

Step-by-Step Solution

1. Volume Calculation for Original Shape:

   • The balloon is made up of a cylindrical part and two hemispheres.
   • Volume of the cylinder: $V_{\text{cyl}} = \pi r^2 h$.
   • Volume of each hemisphere: $V_{\text{hemisphere}} = \frac{2}{3} \pi r^3$.
   • Since there are two hemispheres, the total volume for them is $V_{\text{hemispheres}} = 2 \times \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3$.
   • Total initial volume $V$ of the balloon: $V = V_{\text{cyl}} + V_{\text{hemispheres}} = \pi r^2 h + \frac{4}{3} \pi r^3$

2. Differentiating Volume with Respect to Radius and Height:

   • Let $V$ be a function of $r$ and $h$: $V(r, h) = \pi r^2 h + \frac{4}{3} \pi r^3$.
   • To find the approximate change in volume ($\Delta V$), we can use the multivariable differentiation formula: $\Delta V \approx \frac{\partial V}{\partial r} \Delta r + \frac{\partial V}{\partial h} \Delta h$
   • Compute $\frac{\partial V}{\partial r}$ and $\frac{\partial V}{\partial h}$:
     • $\frac{\partial V}{\partial r} = 2 \pi r h + 4 \pi r^2$
     • $\frac{\partial V}{\partial h} = \pi r^2$

3. Substitute Values:

   • Given $r = 1.5 \, \text{m}$, $h = 4 \, \text{m}$, $\Delta r = 0.01 \, \text{m}$, and $\Delta h = 0.05 \, \text{m}$.
   • Calculate $\Delta V$ using the derived partial derivatives: $\Delta V \approx (2 \pi \cdot 1.5 \cdot 4 + 4 \pi \cdot 1.5^2) \cdot 0.01 + (\pi \cdot 1.5^2) \cdot 0.05$

4. Compute the Percentage Change:

   • Initial volume $V$: $V = \pi \cdot (1.5)^2 \cdot 4 + \frac{4}{3} \pi \cdot (1.5)^3$
   • Percentage change in volume: $\text{Percentage Change} = \left( \frac{\Delta V}{V} \right) \times 100$

By substituting the values and solving, we obtain the percentage change in the balloon’s volume.

Would you like to see each computation step in detail or have further clarification?

Additional Questions to Explore

1. How would the percentage change in volume differ if only the radius increased?
2. What if the balloon were made of a different geometric combination, like a cone with hemispherical ends?
3. How does the change in volume depend on the rate of increase in radius versus height?
4. What is the effect on volume if the radius change were in a negative direction (decreasing radius)?
5. Can this method of approximation be used for larger changes in dimensions?

Tip:

For small changes in dimensions, using partial derivatives is a quick way to approximate the volume change without recalculating the exact volume after each dimension change.