To estimate the maximum error in the radius of a sphere when the circumference is measured, we can use differentials.
The formula for the circumference (C) of a sphere is given by:
C = 2πr
where r is the radius of the sphere. We can rearrange this formula to solve for the radius:
r = C / (2π)
First, we need to find the differential of the circumference:
dC = 2π \, dr
Next, we can relate the possible error in the circumference (dC) to the maximum error in the radius (dr). The possible error in measuring the circumference is given as 0.5 cm, which means:
dC = 0.5 cm
Now we can substitute this into the differential equation to find dr:
0.5 = 2π \, dr
To solve for dr:
dr = 0.5 / (2π)
Now, let’s calculate the value of dr:
dr ≈ 0.5 / 6.2832 ≈ 0.0796 cm (rounded to four decimal places).
Thus, the maximum error in estimating the radius of the sphere is approximately 0.08 cm.