To estimate the maximum error in the radius of a sphere when given its circumference using differentials, we can follow these steps:
The formula for the circumference (C) of a sphere is given by:
C = 2πr
where r is the radius. To find the differential of the circumference, we differentiate both sides with respect to r:
dC = 2π dr
Now, we know that the measurement of the circumference has a potential error. Here, the measured circumference (C) is 84 cm, and the possible error in the measurement (dC) is ±0.5 cm.
Substituting dC into the differential equation, we have:
0.5 = 2π dr
Now we can solve for dr, which represents the change in the radius (the maximum error in the radius):
dr = rac{0.5}{2 ext{π}}
Now, let’s calculate the value of dr:
dr = rac{0.5}{2 imes 3.14} ext{ (approximately)}
dr ≈ rac{0.5}{6.28} ≈ 0.0796 ext{ cm}
This means that the maximum error in the radius of the sphere is approximately ±0.0796 cm.
In conclusion, when measuring the circumference of the sphere to be 84 cm with an error of 0.5 cm, we can estimate the maximum possible error in the radius using differentials to be about ±0.0796 cm.