To estimate the maximum error in the calculated surface area of the sphere, we can use linear approximation.
The surface area of a sphere can be calculated using the formula A = 4Ï€r^2, where A is the surface area and r is the radius of the sphere.
First, let's find the measured radius of the sphere using the circumference. The formula for circumference is C = 2Ï€r, where C is the circumference and r is the radius.
Given that the circumference is 74 cm with a possible error of 0.5 cm, we can write this as:
C = 74 cm ± 0.5 cm
Now, solve for the radius:
C = 2Ï€r
74 cm ± 0.5 cm = 2πr
To estimate the maximum error in the calculation of the surface area, we need to find the maximum and minimum values for the radius.
For the maximum value of the radius, add the maximum error to the measured value of the circumference:
74 cm + 0.5 cm = 74.5 cm
For the minimum value of the radius, subtract the maximum error from the measured value of the circumference:
74 cm - 0.5 cm = 73.5 cm
Now we will calculate the surface areas for both the maximum and minimum values of the radius and determine the difference:
Maximum surface area:
A_max = 4Ï€(74.5 cm)^2
Minimum surface area:
A_min = 4Ï€(73.5 cm)^2
Now, find the difference between the maximum surface area and the minimum surface area:
ΔA = A_max - A_min
This difference will give us the maximum error in the calculated surface area.
I will now calculate the values for you.