Understanding the Relationship Between Volume and Surface Area
The volume of a sphere is given by the formula V = (4/3)πr³, while the surface area is given by the formula A = 4πr². To find the error in surface area based on the error in volume, we need to understand how these two quantities relate through the radius (r).
Percentage Error in Volume
- The problem states that the error in the volume measurement is 6%.
- This means that if the actual volume is V, the measured volume V' can be expressed as V' = V ± 0.06V.
Calculating Percentage Error in Radius
- The volume of the sphere is proportional to the cube of the radius: V ∝ r³.
- Therefore, the relative error in volume (percentage error) leads to a relative error in radius.
- Using the relation, if the volume has a 6% error, the error in the radius can be determined by the formula:
Percentage Error in Radius = (1/3) * Percentage Error in Volume
- Thus, the error in radius = (1/3) * 6% = 2%.
Calculating Surface Area Error
- The surface area is proportional to the square of the radius: A ∝ r².
- The percentage error in surface area can be derived from the error in radius:
Percentage Error in Surface Area = 2 * Percentage Error in Radius
- Hence, the error in surface area = 2 * 2% = 4%.
Conclusion
- Therefore, if the error in the measurement of the volume of a sphere is 6%, the error in the measurement of the surface area will be 4%. This relationship illustrates how changes in one geometric property influence others through their mathematical relationships.