Math Problem Statement

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.)

Solution

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)=43πr3V(r) = \frac{4}{3} \pi r^3, where rr is the radius.

Step-by-step approach:

  1. Volume formula for a sphere: V(r)=43πr3V(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, dVdV, gives an approximation of the change in volume for a small change in radius drdr: dV=ddr(43πr3)dr=4πr2drdV = \frac{d}{dr} \left( \frac{4}{3} \pi r^3 \right) \cdot dr = 4 \pi r^2 \cdot dr Here, r=5r = 5 mm and dr=5.25=0.2dr = 5.2 - 5 = 0.2 mm.

  3. Substitute into the differential equation: dV=4π(5)20.2=4π250.2=20πmm3dV = 4 \pi (5)^2 \cdot 0.2 = 4 \pi \cdot 25 \cdot 0.2 = 20 \pi \, \text{mm}^3

  4. Calculate the volume change: dV20×3.1416=62.83mm3dV \approx 20 \times 3.1416 = 62.83 \, \text{mm}^3

Thus, the approximate volume of the shell is 62.83mm362.83 \, \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!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Differentials
Geometry
Volume of a Sphere

Formulas

V(r) = (4/3)πr³
dV = 4πr²dr

Theorems

Differentiation of Volume Formula

Suitable Grade Level

Grades 10-12