he radius of a circular disk is given as 19 cm with a maximum error in measurement of 0.2 cm.(a) Use differentials to estimate the maximum error in the calculated area of the disk. (Round your answer to two decimal places.) cm2(b) What is the relative error? (Round your answer to four decimal places.)What is the percentage error? (Round your answer to two decimal places.)
Question
he radius of a circular disk is given as 19 cm with a maximum error in measurement of 0.2 cm.(a) Use differentials to estimate the maximum error in the calculated area of the disk. (Round your answer to two decimal places.) cm2(b) What is the relative error? (Round your answer to four decimal places.)What is the percentage error? (Round your answer to two decimal places.)
Solution
(a) The formula for the area of a circle is A = πr². We can find the differential of this function, dA, to estimate the maximum error in the calculated area of the disk.
dA = d(πr²) = 2πr dr
Given that the radius r = 19 cm and the maximum error in the radius dr = 0.2 cm, we can substitute these values into the equation:
dA = 2π(19 cm)(0.2 cm) = 23.76 cm²
So, the maximum error in the calculated area of the disk is approximately 23.76 cm².
(b) The relative error is the absolute error divided by the measured value. In this case, it's the maximum error in the area divided by the actual area of the disk.
Relative error = dA / A = 23.76 cm² / (π(19 cm)²) = 0.0263
So, the relative error is approximately 0.0263 or 0.0263 when rounded to four decimal places.
The percentage error is the relative error expressed as a percentage.
Percentage error = Relative error x 100% = 0.0263 x 100% = 2.63%
So, the percentage error is approximately 2.63%.
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