he circumference of a sphere was measured to be 78000 cm with a possible error of 050000 cm. Use linear approximation to estimate the maximum error in the calculated surface area.
To estimate the maximum error in the calculated surface area of the sphere, we can use linear approximation.
1. Start by finding the radius of the sphere. The formula for circumference is C = 2πr, so we can rearrange it to solve for the radius, r = C / (2π).
In this case, the circumference is given as 78000 cm. So, the radius would be r = 78000 / (2π) ≈ 12394.44 cm.
2. Next, calculate the surface area of the sphere using the formula A = 4πr².
Let's find the surface area with the measured radius first: A = 4π(12394.44)² ≈ 1925790710.88 cm².
3. Now, let's determine the maximum error in the radius. The possible error is given as 50000 cm.
Since radius is directly related to the circumference, we can approximate the maximum error in the radius as Δr = ΔC / (2π).
In this case, the possible error in circumference is 50000 cm, so the maximum error in the radius would be Δr = 50000 / (2π) ≈ 7957.75 cm.
4. Using the maximum error in the radius, we can estimate the maximum error in the calculated surface area.
The formula to estimate the maximum error in a function using linear approximation is given by Δf ≈ |f'(x)| * Δx, where f'(x) is the derivative of the function and Δx is the maximum error in x.
In this case, we need to find the derivative of the surface area function, A' = dA/dr = 8πr.
5. Finally, plug in the values into the formula to get the estimated maximum error in the surface area:
ΔA ≈ |8πr * Δr| = |8π * 12394.44 * 7957.75| ≈ 2476122013.71 cm².
Therefore, using linear approximation, the estimated maximum error in the calculated surface area of the sphere is approximately 2476122013.71 cm².