The circumference of a sphere was measured to be 89.000 cm with a possible error of 0.50000 cm.
A. Use linear approximation to estimate the maximum error in the calculated surface area.
B. Estimate the relative error in the calculated surface area.
since
v = 4/3 pi r^3
and c = 2pi r, r = c/2pi, and
v = 4/3 pi (c/2pi)^3
v = 1/(6pi^2) c^3
so, dv = 1/(2pi^2) c^2 dc
so, now you have dc and can evaluate dv
A. To estimate the maximum error in the calculated surface area, we first need to find the derivative of the surface area formula with respect to the circumference. The surface area of a sphere is given by the formula:
S = 4πr²
where S is the surface area and r is the radius of the sphere.
The circumference of a sphere is related to the radius by the formula:
C = 2πr
We can rewrite the surface area formula in terms of the circumference by substituting the radius from the circumference formula:
S = π(C/2π)²
= πC²/4π²
= C²/4π
Now, let's find the derivative of S with respect to C:
dS/dC = (2C)/(4π)
= C/(2π)
To find the maximum error in the surface area calculation, we need to multiply the maximum error in the circumference measurement by the absolute value of the derivative:
Maximum error in surface area = |dS/dC| * maximum error in circumference measurement
Given that the maximum error in the circumference measurement is 0.50000 cm, we can substitute this value into the equation:
Maximum error in surface area = (0.50000 cm) * (C/(2π))
= 0.50000C/(2π)
Since the measured circumference is 89.000 cm, we can substitute this value for C:
Maximum error in surface area = 0.50000 * (89.000 cm)/(2π)
Calculating this expression will give us the estimated maximum error in the calculated surface area.
B. To estimate the relative error in the calculated surface area, we can divide the maximum error in surface area by the actual surface area. Let's denote the relative error as E:
E = (Maximum error in surface area) / (Actual surface area)
The actual surface area can be calculated using the measured circumference:
Actual surface area = 4πr²
= 4π((C/2π) / (2π))^2
= 4π(C²/4π²)
= C²/π
Substituting the values, the relative error can be calculated as:
Relative error = (0.50000C/(2π)) / (C²/π)
= (0.50000C/(2π)) * (π/C²)
= 0.50000 / (2C)
Again, substituting the measured circumference value:
Relative error = 0.50000 / (2 * 89.000 cm)