The circumference of a sphere was measured to be 83000 cm with a possible error of 050000 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, we can use linear approximation.
The formula for the surface area of a sphere is given by A = 4πr^2, where r is the radius of the sphere.
Since the circumference of the sphere is given as 83000 cm with a possible error of 050000 cm, we can use this value to find the radius. The formula for the circumference of a sphere is C = 2πr.
83000 cm = 2πr
Dividing both sides by 2π, we get:
r = 83000 cm / (2π)
To calculate the maximum error in the surface area, we need to find the derivative of the surface area formula with respect to r and multiply it by the maximum error in the radius.
The derivative of A = 4πr^2 with respect to r is:
dA/dr = 8πr
To estimate the maximum error in the surface area, we can use the formula for linear approximation:
ΔA ≈ |dA/dr| * Δr
where ΔA is the maximum error in the surface area and Δr is the maximum error in the radius.
Substituting the derived values, we have:
ΔA ≈ |8πr| * 050000 cm
Now, let's calculate the maximum error in the surface area using the estimated radius.
r = 83000 cm / (2π) ≈ 83000 cm / 6.28 ≈ 13216.56 cm
ΔA ≈ |8π(13216.56 cm)| * 050000 cm
≈ 8 * 3.14 * 13216.56 cm * 050000 cm
≈ 41759104.64 cm^2
Therefore, the estimated maximum error in the calculated surface area is approximately 41759104.64 cm^2.
To find the relative error in the calculated surface area, we need to divide the estimated maximum error by the actual surface area.
The actual surface area of the sphere is given by S = 4πr^2, where r = 13216.56 cm.
S = 4π(13216.56 cm)^2
= 4 * 3.14 * (13216.56 cm)^2
≈ 2,199,087,831.09 cm^2
Relative error = (estimated maximum error / actual surface area) * 100
= (41759104.64 cm^2 / 2,199,087,831.09 cm^2) * 100
≈ 1.90%
Therefore, the estimated relative error in the calculated surface area is approximately 1.90%.
To estimate the maximum error in the calculated surface area of the sphere, we need to use linear approximation. Here's how we can do it:
1. Recall that the surface area (A) of a sphere is given by the formula: A = 4πr^2, where r is the radius of the sphere.
2. We are given the circumference (C) of the sphere, which is related to the radius (r) by the formula: C = 2πr. Solving this equation for r, we find that r = C/(2π).
3. The maximum error in the circumference is given as 050000 cm. This means the actual circumference could be 050000 cm larger or smaller than the measured value of 83000 cm. So, the range of possible circumferences is from 83000 cm - 050000 cm to 83000 cm + 050000 cm, or from -170000 cm to 133000 cm.
4. Using these range values, we can calculate the range of possible radii by dividing the range of circumferences by 2π. Therefore, the range of possible radii is from -170000 cm divided by 2π to 133000 cm divided by 2π.
5. To find the maximum error in the calculated surface area, we'll use linear approximation. The formula for the linear approximation of a function f(x) is: Δf ≈ f'(x) * Δx, where Δf is the change in the function, f'(x) is the derivative of the function, and Δx is the change in the input.
6. Taking the derivative of the surface area function with respect to the radius, we get: dA/dr = 8πr.
7. Since we are interested in the maximum error, we'll take the absolute value and multiply it by the maximum error in the radius. The maximum error in the surface area is then given by: ΔA_max ≈ |dA/dr| * Δr_max.
8. Plugging in the values, we have: Δr_max = (133000 cm / 2π) - (-170000 cm / 2π) and |dA/dr| = |8πr|.
9. After calculating the values, we find Δr_max ≈ 51646.3171 cm and |dA/dr| ≈ 8πr ≈ 8π(83000 cm / (2π)) = 332000 cm.
10. Finally, we can calculate the maximum error in the surface area using the formula: ΔA_max ≈ |dA/dr| * Δr_max. Plugging in the values, we get ΔA_max ≈ 332000 cm * 51646.3171 cm≈ 1.713 * 10^10 cm^2.
Therefore, the maximum error in the calculated surface area of the sphere is approximately 1.713 * 10^10 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 radius obtained from the linear approximation.
1. We obtained the radius value from step 3, which is 133000 cm / 2π.
2. We can now substitute this radius value into the surface area formula (A = 4πr^2) to find the actual surface area of the sphere.
3. Once we have the actual surface area, we can calculate the relative error by dividing the maximum error in the surface area by the actual surface area and multiplying by 100 to express it as a percentage.
Therefore, by following the steps above, we can estimate the relative error in the calculated surface area of the sphere.