The circumference of a sphere was measured to be 81000 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 a sphere, we can use linear approximation by considering the derivatives of the formulas for circumference and surface area of a sphere.
The formula for the circumference of a sphere is:
C = 2πr
The formula for the surface area of a sphere is:
A = 4πr^2
Let's first find the derivative of the formula for circumference with respect to the radius:
dC/dr = 2π
This tells us that if there is a small change Δr in the radius, the change ΔC in the circumference is approximately ΔC ≈ 2πΔr.
Similarly, let's find the derivative of the formula for surface area with respect to the radius:
dA/dr = 8πr
This tells us that if there is a small change Δr in the radius, the change ΔA in the surface area is approximately ΔA ≈ 8πrΔr.
Now, let's use the given values to estimate the maximum error in the calculated surface area.
The measured circumference of the sphere was 81000 cm with a possible error of 5000 cm. So, the actual circumference could range from 81000 - 5000 = 76000 cm to 81000 + 5000 = 86000 cm.
We want to estimate the maximum error in the calculated surface area, so let's use the maximum value of the circumference (86000 cm).
Using the formula for circumference, we can solve for the radius:
C = 2πr
86000 = 2πr
r ≈ 86000 / (2π) ≈ 13680.27 cm
Now, let's find the maximum error in the radius due to the possible error in the circumference:
Δr = (maximum circumference error) / (2π) = (5000 cm) / (2π) ≈ 795.77 cm
Using the derivative of the surface area formula, we can estimate the maximum error in the surface area:
ΔA ≈ 8πrΔr ≈ 8π(13680.27 cm)(795.77 cm) ≈ 2741710.52 cm²
Therefore, the maximum error in the calculated surface area is approximately 2741710.52 cm².