Estimating Maximum Error in Surface Area
To determine the maximum error in the surface area of a sphere, we can use differentials. First, we need to establish a relationship between the measurements of the circumference and the surface area.
Step 1: Formulas
The circumference (C) of a sphere is given by the formula:
C = 2πr
Where r is the radius of the sphere.
The surface area (A) of a sphere is given by the formula:
A = 4πr²
Step 2: Expressing the Radius in Terms of Circumference
From the circumference formula, we can solve for the radius:
r = C / (2π)
Step 3: Finding the Differential of Surface Area
To find the differential of the surface area (dA), we first need to find the differential of the radius (dr) using the circumference:
By differentiating the circumference formula, we have:
dC = 2πdr
So, we can express dr in terms of dC:
dr = dC / (2π)
Step 4: Finding dA in Terms of dr
Next, we differentiate the surface area formula:
dA = 8πr dr
Step 5: Substituting the Variables
Now, we substitute dr into this equation:
dA = 8πr (dC / (2π)) = 4r dC
Step 6: Calculate the Values
The measured circumference is C = 74 cm with a possible error of dC = 0.5 cm.
First, we need to calculate the radius:
r = 74 / (2π) ≈ 11.77 cm
Now, substituting the value of r and dC into the dA equation:
dA = 4 * 11.77 * 0.5 ≈ 11.77 cm²
Final Result
Thus, the maximum error in the calculated surface area of the sphere is approximately 11.77 cm².