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The problem describes an egg that is nearly spherical. The radius to the inside of the shell is 5 millimeters, and the radius to the outside of the shell is 5.2 millimeters. We are asked to use differentials to approximate the volume of the shell, given that the volume of a sphere is \( V(r) = \frac{4}{3} \pi r^3 \), where \( r \) is the radius.

### Step-by-step approach:

1. **Volume formula for a sphere:**
   \[
   V(r) = \frac{4}{3} \pi r^3
   \]
   We need to approximate the volume of the shell, which is the difference between the volume of the outer sphere (radius 5.2 mm) and the inner sphere (radius 5 mm).

2. **Using differentials to approximate the change in volume:**
   The differential of volume, \( dV \), gives an approximation of the change in volume for a small change in radius \( dr \):
   \[
   dV = \frac{d}{dr} \left( \frac{4}{3} \pi r^3 \right) \cdot dr = 4 \pi r^2 \cdot dr
   \]
   Here, \( r = 5 \) mm and \( dr = 5.2 - 5 = 0.2 \) mm.

3. **Substitute into the differential equation:**
   \[
   dV = 4 \pi (5)^2 \cdot 0.2 = 4 \pi \cdot 25 \cdot 0.2 = 20 \pi \, \text{mm}^3
   \]

4. **Calculate the volume change:**
   \[
   dV \approx 20 \times 3.1416 = 62.83 \, \text{mm}^3
   \]

Thus, the approximate volume of the shell is \( 62.83 \, \text{mm}^3 \).

Let me know if you need further clarification or details!

Here are 5 related questions for further practice:
1. What would be the volume of the shell if the radii were 6 mm and 6.2 mm?
2. How would the volume of the shell change if the thickness of the shell was doubled?
3. What is the exact volume of the shell without using differentials?
4. How does the approximation change if we used a larger radius for the inner shell?
5. Can you derive the formula for the volume of a shell directly using integration?

**Tip:** Always remember that using differentials provides an approximation, which is most accurate when the change in the radius is small!

An egg of a particular bird is very nearly spherical. The radius to the inside of the shell is 5 millimeters and the radius to the outside of the shell is 5.2 millimeters. Use differentials to approximate the volume of the shell. (Remember that V(r) = (4/3)πr³, where r is the radius.)

Learn how to use differentials to approximate the volume of a spherical egg shell with an inner radius of 5 mm and an outer radius of 5.2 mm. This problem involves the formula V(r) = (4/3)πr³ and is suitable for high school students in Grades 10-12.

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