The circumference of a sphere was measured to be 81 cm with a possible error of 0.5 cm. Use linear approximation to estimate the maximum error in the calculated surface area
sarea= 4PI r^2
errorArea= 8PI r dr
so the maximum error is +- 8*PI*81*.5
To estimate the maximum error in the calculated surface area of a sphere, we can use linear approximation.
Step 1: Start by finding the radius of the sphere. The formula for the circumference of a sphere is given by C = 2πr, where C is the circumference and r is the radius. Rearranging the formula, we have r = C / (2π).
Given that the circumference (C) of the sphere is 81 cm, we can calculate the radius (r) using the formula:
r = 81 cm / (2π) ≈ 12.888 cm (rounded to three decimal places)
Step 2: Calculate the surface area of the sphere using the formula A = 4πr^2, where A is the surface area of the sphere.
A = 4π(12.888 cm)^2
≈ 2078.435 cm² (rounded to three decimal places)
So, the calculated surface area of the sphere is approximately 2078.435 cm².
Step 3: Determine the maximum error in the circumference. The possible error in the circumference is given as 0.5 cm.
Therefore, the maximum error in the radius is equal to half of the maximum error in the circumference.
Maximum error in radius (δr) = 0.5 cm / (2π) ≈ 0.0796 cm (rounded to four decimal places)
Step 4: Estimate the maximum error in the surface area using linear approximation. The formula for linear approximation is given by:
δA = 2πrδr
Substituting the values we have:
δA = 2π(12.888 cm)(0.0796 cm)
≈ 6.418 cm² (rounded to three decimal places)
Therefore, the maximum error in the calculated surface area of the sphere is approximately 6.418 cm².