This problem involves determining the potential variation in the volume of a sphere when there's uncertainty in its radius measurement. We are given a sphere with a measured radius of 10 mm, and a possible error of ±0.1 mm. Our goal is to calculate the maximum possible change in the volume of the sphere....
The Formula for Sphere Volume
The volume (V) of a sphere is calculated using the following formula:
V = (4/3)πr³
where:
- V is the volume of the sphere
- π (pi) is a mathematical constant approximately equal to 3.14159
- r is the radius of the sphere
Calculating the Volume with the Measured Radius
First, we need to calculate the volume of the sphere using the given measured radius of 10 mm:
V = (4/3)π(10 mm)³
V ≈ 4188.79 mm³
Calculating the Maximum and Minimum Volumes
Now, we consider the possible error in the radius measurement. To find the maximum possible volume, we'll use the largest possible radius, which is 10 mm + 0.1 mm = 10.1 mm.
V_max = (4/3)π(10.1 mm)³
V_max ≈ 4300.28 mm³
For the minimum possible volume, we'll use the smallest possible radius, which is 10 mm - 0.1 mm = 9.9 mm.
V_min = (4/3)π(9.9 mm)³
V_min ≈ 4080.64 mm³
Determining the Possible Change in Volume
To find the possible change in volume, we need to calculate the difference between the maximum and minimum volumes:
ΔV = V_max - V_min
ΔV ≈ 4300.28 mm³ - 4080.64 mm³
ΔV ≈ 219.64 mm³
Conclusion
Therefore, the possible change in volume of the sphere due to the measurement error in the radius is approximately ±219.64 mm³. This means the actual volume of the sphere could be anywhere between 4080.64 mm³ and 4300.28 mm³.
Importance of Measurement Error
This example highlights the significance of considering measurement errors in calculations. Even a small error in the radius measurement can result in a considerable difference in the calculated volume, especially when dealing with three-dimensional objects.
Applications in Real-World Scenarios
This concept has numerous applications in various fields, including:
- Engineering: Accurately calculating the volume of materials used in construction, manufacturing, and other engineering projects.
- Manufacturing: Ensuring the consistency of product dimensions and volumes.
- Science: Analyzing and interpreting data collected from experiments involving measurements.
- Medicine: Determining the appropriate dosages of medication based on patient measurements.
Further Considerations
It's important to note that the possible change in volume calculated above represents the maximum potential variation. In reality, the actual error in volume might be smaller. However, it's crucial to consider the worst-case scenario to ensure accurate and reliable results.
Furthermore, when dealing with measurement errors, it's beneficial to use methods like error propagation to quantify the impact of measurement uncertainties on the calculated values. This helps in obtaining more accurate estimations and understanding the reliability of results.