If the radius of a sphere is measured as 7 m with an error of 0.02m, then find the approximate error in calculating its volume.
Solution and Explanation
Let r be the radius of the sphere and ∆r be the error in measuring the radius. Then, r = 7 m and ∆r = 0.02 m Now, the volume V of the sphere is given by,
v=\(\frac{4}{3}\)πr3
\(\Rightarrow\)\(\frac{dv}{dr}\)=4πr2
\(\Rightarrow\) dv=(\(\frac{dv}{dr}\))∇r
=(4πr2)∇r
4π(7)2(0.02)m3=3.92πm3
Hence, the approximate error in calculating the volume is 3.92 πm3.
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Notes on Applications of Derivatives
Concepts Used:
Errors in Measurement
Errors in measurement refer to the differences between the true value of a quantity and the value obtained from a measurement. Measurement errors can arise from a variety of sources, including limitations of the measuring instrument, the measuring technique, and the observer performing the measurement.
There are two types of errors in measurement: systematic errors and random errors. Systematic errors are caused by a flaw in the measurement system or method that consistently leads to a deviation from the true value. Random errors, on the other hand, arise from unpredictable and uncontrollable factors and cause fluctuations in the measured values.
There are several sources of systematic errors, such as calibration errors, instrument drift, parallax errors, and environmental conditions. Calibration errors occur when the measuring instrument is not calibrated correctly, leading to incorrect measurements. Instrument drift refers to a gradual change in the measuring instrument's characteristics, which can cause measurements to be consistently inaccurate over time. Parallax errors occur when the observer's eye is not aligned correctly with the measuring instrument, leading to an error in the reading. Environmental conditions, such as temperature, humidity, and pressure, can also cause systematic errors.
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Random errors can be caused by a variety of factors, including variations in the measurement technique, inherent variability in the quantity being measured, and fluctuations in environmental conditions. These errors can be reduced by taking multiple measurements and calculating an average value.
It is important to understand and minimize measurement errors to ensure accurate and reliable data. Calibration of instruments, careful observation, and consistent measurement techniques can help reduce errors and improve the quality of measurements.