The problem asks us to approximate the volume of the shell of a bird's egg, which is nearly spherical. It provides the following information:

• The radius to the inside of the shell is $r = 5$ mm.
• The radius to the outside of the shell is $r + \Delta r = 5.3$ mm.
• The formula for the volume of a sphere is $V(r) = \frac{4}{3} \pi r^3$.

Step 1: Differential Approximation of Volume

We want to use differentials to approximate the volume of the shell, which is the difference between the volume of the outer sphere and the inner sphere.

The differential of volume with respect to $r$ is: $dV = \frac{d}{dr} \left( \frac{4}{3} \pi r^3 \right) = 4 \pi r^2 \, dr$

Where $dr$ is the change in radius, i.e. $\Delta r = 5.3 - 5 = 0.3$ mm.

Step 2: Approximate Change in Volume

Now, to approximate the volume of the shell, we compute the differential at $r = 5$ mm with $dr = 0.3$ mm: $dV = 4 \pi (5)^2 \times 0.3$ $dV = 4 \pi \times 25 \times 0.3 = 30 \pi \, \text{mm}^3$

Thus, the approximate volume of the shell is $30 \pi \, \text{mm}^3$, which is about: $dV \approx 94.25 \, \text{mm}^3$

This is the approximate volume of the bird's eggshell using differentials.

Let me know if you want any further details or clarifications. Here are some related questions:

1. How would the result change if the thickness of the shell was doubled?
2. Can you derive the volume of the shell using exact calculations instead of differentials?
3. How would we calculate the volume for an egg-shaped object, which is not perfectly spherical?
4. What is the geometric significance of the differential approximation in this context?
5. How does using differentials simplify the calculation of small changes in volume?

Tip: Using differentials is a powerful tool for estimating small changes in quantities when dealing with smooth functions like the volume of a sphere.