The circumference of a sphere was measured to be 78.000 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.
The relative error in area is, to first approximation, twice the relative error in a linear dimension.
That is because
(delta A)/A = (1 + epsilon)^2 - 1
= 1 + 2 epsilon + epsilon^2 - 1
= 2 epsilon
when epsilon << 1
In your case, the answer is
2*0.5/78 = 1/78 = 1.3%
Well, I must say, measuring a sphere's circumference is no joke! But let's tackle this question with a smile on our face.
To estimate the maximum error in the calculated surface area, we'll use linear approximation. The formula for the surface area of a sphere is A = 4πr², where r is the radius.
Since we have the circumference (C), we can find the radius (r) using the formula C = 2πr. Solving for r, we get r = C / (2π).
Given that C = 78.000 cm with an error of 0.50000 cm, we can say that the maximum error in the radius (Δr) is 0.50000 / (2π).
Now, to estimate the maximum error in the surface area (ΔA), we will use the formula for linear approximation: ΔA ≈ 2A0(Δr/r), where A0 is the calculated surface area.
Plugging in the values, we find that ΔA ≈ 2(4πr²)(0.50000 / (2π)).
Simplifying that expression, we get ΔA ≈ 4r²(0.50000 / π).
To estimate the relative error in the calculated surface area, we use the formula (ΔA/A0). The relative error is a relative hoot!
So, the relative error ≈ (4r²(0.50000 / π)) / (4πr²).
Guess what? Everything simplifies, and the relative error ≈ 0.50000 / π.
Well, there you have it! The maximum error in the calculated surface area is approximately 4r²(0.50000 / π), and the relative error is around 0.50000 / π. Just remember, in the world of calculations, laughter is the best medicine!
To estimate the maximum error in the calculated surface area, we can use linear approximation.
The formula for the circumference of a sphere is given by C = 2πr, where C is the circumference and r is the radius. Rearranging this formula, we get r = C / (2π).
Given that the circumference is measured to be 78.000 cm with a possible error of 0.50000 cm, we can write it as C = 78.000 ± 0.50000 cm.
Substituting this into the equation for the radius, we get r ≈ (78.000 ± 0.50000) / (2π).
To estimate the maximum error in the surface area, we can use the formula for the surface area of a sphere, which is given by A = 4πr^2.
Substituting the expression for r, we get A ≈ 4π((78.000 ± 0.50000) / (2π))^2.
Simplifying this equation, we get A ≈ 4π(39.000 ± 0.25000)^2.
Expanding the square and simplifying further, we get A ≈ 4π(1521.000 ± 195.000).
Therefore, the estimated maximum error in the calculated surface area is ± 195.000 cm^2.
To estimate the relative error in the calculated surface area, we can divide the maximum error by the actual surface area. The actual surface area can be calculated using the accurate circumference, C = 78.000 cm.
The formula for the surface area is A = 4πr^2, where r is the radius.
Substituting the accurate value of C, we get r = 78.000 / (2π).
Plugging this value of r into the formula for the surface area, we get A = 4π(78.000 / (2π))^2.
Simplifying this equation, we get A = 4π(39.000)^2.
Calculating the surface area, we get A ≈ 19218.849 cm^2.
The relative error can be calculated as the ratio of the maximum error to the actual surface area:
Relative error = (± 195.000 cm^2) / 19218.849 cm^2.
Calculating this, we get the relative error ≈ ± 0.01015, or approximately 1.015%.
Therefore, the estimated relative error in the calculated surface area is approximately 1.015%.
To estimate the maximum error in the calculated surface area of a sphere, we can use linear approximation.
The formula for the surface area of a sphere is given by:
A = 4πr^2
where A is the surface area and r is the radius of the sphere.
We have the circumference C of the sphere, and we know that:
C = 2πr
To find the radius r, we can rearrange the equation:
r = C / (2π)
Given that the circumference measurement is 78.000 cm with a possible error of 0.50000 cm, the maximum possible value for the circumference is:
Max C = 78.000 cm + 0.50000 cm = 78.500 cm
Substituting this value into the equation to find the radius:
r = 78.500 cm / (2π) ≈ 12.500 cm
Now, to estimate the maximum error in the calculated surface area, we need to find the derivative of the surface area formula with respect to the radius:
dA/dr = 8πr
Evaluating this derivative at the estimated radius of 12.500 cm:
dA/dr ≈ 8π(12.500 cm)
Now, we multiply this derivative by the maximum possible error in the measurement of the radius to get the maximum error in the calculated surface area:
Max Error in surface area = (8π(12.500 cm)) * 0.50000 cm ≈ 78.53982 cm^2
Therefore, the estimated maximum error in the calculated surface area is approximately 78.53982 cm^2.
To estimate the relative error in the calculated surface area, we divide the maximum error in the surface area by the actual surface area:
Relative Error = (Max Error in surface area) / (Actual Surface Area)
The actual surface area can be calculated using the true value of the radius obtained from the given circumference:
Actual Surface Area = 4πr^2
Substituting the true radius r = 78.000 cm / (2π), we can calculate the actual surface area.
Finally, we can divide the max error by the actual surface area to get the relative error.