The circumference of a sphere was measured to be 76.000 cm with a possible error of 0.50000 cm. Use linear approximation to estimate the maximum error in the calculated surface area.Estimate the relative error in the calculated surface area.
To estimate the maximum error in the calculated surface area of the sphere, we'll use linear approximation. The linear approximation, also known as the differential approximation, is based on the fact that for small changes in a function, the change in the function is approximately proportional to the change in its independent variable.
Let's denote the radius of the sphere as r and the circumference as C. We have the formula for the circumference of a sphere: C = 2πr.
Given that C = 76.000 cm with a possible error of 0.50000 cm, we can express this as C ± δC, where δC = 0.50000 cm is the maximum error.
Now, let's differentiate both sides of the equation C = 2πr with respect to r:
dC = 2πdr,
where dC represents the change in the circumference and dr represents the change in the radius.
Approximating the change in circumference as the maximum error δC and the change in radius as the maximum error δr, we have:
δC ≈ 2πδr.
Now, we can rearrange this equation to solve for δr:
δr ≈ δC / (2π).
Substituting the maximum error values, we have:
δr ≈ 0.50000 cm / (2π).
Calculating this value, we find:
δr ≈ 0.079577 cm.
So, the estimated maximum error in the radius is approximately 0.079577 cm.
Next, we need to estimate the relative error in the calculated surface area. The surface area of a sphere is given by the formula: A = 4πr^2.
Differentiating both sides with respect to r gives:
dA = 8πrdr.
Again, approximating the change in surface area as the maximum error δA and the change in radius as the maximum error δr, we have:
δA ≈ 8πrδr.
Substituting the known values, we get:
δA ≈ 8π(76.000 cm)(0.079577 cm).
Calculating this value, we find:
δA ≈ 151.424 cm².
So, the estimated maximum error in the calculated surface area of the sphere is approximately 151.424 cm².
To estimate the maximum error in the calculated surface area, we can use linear approximation. The surface area formula for a sphere is given by:
A = 4πr^2
Where A is the surface area and r is the radius of the sphere.
Given that the circumference of the sphere is 76.000 cm with a possible error of 0.50000 cm, we can use this information to find the radius.
The circumference formula for a sphere is given by:
C = 2πr
Where C is the circumference and r is the radius.
Let's solve the equation for the radius:
2πr = 76.000 cm
r = 76.000 cm / (2π)
r ≈ 12.080 cm
Now, let's calculate the surface area using the estimated radius:
A ≈ 4π(12.080)^2
A ≈ 1831.711 cm^2
To estimate the maximum error in the calculated surface area, we can differentiate the surface area formula with respect to the radius:
dA/dr = 8πr
Now substitute the value of the estimated radius:
dA/dr ≈ 8π(12.080)
dA/dr ≈ 303.243 cm^2/cm
The maximum error in the calculated surface area can be found by multiplying the maximum error in the radius by the derivative of the surface area formula:
Maximum error in surface area = (0.50000 cm) * (303.243 cm^2/cm)
Maximum error in surface area ≈ 151.621 cm^2
To estimate the relative error in the calculated surface area, we can divide the maximum error in the surface area by the calculated surface area:
Relative error in surface area = (151.621 cm^2) / (1831.711 cm^2)
Relative error in surface area ≈ 0.0827 or 8.27% (rounded to four decimal places)