Math Problem Statement
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 , where is the radius.
Step-by-step approach:
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Volume formula for a sphere: 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).
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Using differentials to approximate the change in volume: The differential of volume, , gives an approximation of the change in volume for a small change in radius : Here, mm and mm.
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Substitute into the differential equation:
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Calculate the volume change:
Thus, the approximate volume of the shell is .
Let me know if you need further clarification or details!
Here are 5 related questions for further practice:
- What would be the volume of the shell if the radii were 6 mm and 6.2 mm?
- How would the volume of the shell change if the thickness of the shell was doubled?
- What is the exact volume of the shell without using differentials?
- How does the approximation change if we used a larger radius for the inner shell?
- 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!
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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
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