The circumference of a sphere was measured to be 86.000 cm with a possible error of 0.5 cm. Use linear approximation to estimate the maximum error in the calculated surface area and Estimate the relative error in the calculated surface area.
To estimate the maximum error in the calculated surface area, we can use linear approximation.
The formula for the circumference of a sphere is given by C = 2πr, where C is the circumference and r is the radius.
In this case, the circumference is measured to be 86.000 cm with a possible error of 0.5 cm. This means that the actual circumference could be anywhere between 86.000 cm + 0.5 cm = 86.500 cm and 86.000 cm - 0.5 cm = 85.500 cm.
Let's find the radius using the measured circumference of 86.000 cm:
C = 2πr
86.000 = 2πr
Dividing both sides by 2π, we get:
r = 86.000 / (2π)
Using a calculator, we find that r ≈ 13.677 cm (rounded to three decimal places).
Now, let's find the maximum error in the calculated surface area. The formula for the surface area of a sphere is given by A = 4πr^2.
Let A1 be the calculated surface area using the measured radius of 13.677 cm, and A2 be the calculated surface area using the maximum possible radius of 13.677 cm + 0.5 cm = 14.177 cm.
A1 = 4π(13.677)^2
A2 = 4π(14.177)^2
Using a calculator, we find that A1 ≈ 2951.132 cm² and A2 ≈ 3164.844 cm².
The maximum error in the calculated surface area is given by the difference between A2 and A1:
Maximum error = A2 - A1 ≈ 3164.844 cm² - 2951.132 cm² ≈ 213.712 cm² (rounded to three decimal places).
To estimate the relative error in the calculated surface area, we can divide the maximum error by the actual surface area (A1):
Relative error = (Maximum error / A1) * 100%
Relative error ≈ (213.712 cm² / 2951.132 cm²) * 100% ≈ 7.241% (rounded to three decimal places).
Therefore, the estimated maximum error in the calculated surface area is approximately 213.712 cm², and the estimated relative error is approximately 7.241%.