You need to remember the formula of the circumference of circle, hence you need to use this formula to find the radius of sphere such that:

`C= 2pi*r =gt r = C/(2pi)`

You need to evaluate the surface area of sphere such that:

`A = 4pi*r^2`

Substituting `C/(2pi)` for r yields:

`A = 4pi*C^2/(4pi^2)`

Reducing like terms yields:

`A = (C^2)/pi`

You need to remember that linear approximation means to use the tangent line to the given function at a point to find the approximate value of the function around that point.

Hence, you need to differentiate both sides the formula of area such that:

`dA = ((2C)/pi) dC`

Substituting 82.000 for C and 0.5 for dC yields:

`dA = (164.000/pi)*0.5`

`dA = 26101.410 `

**Hence, evaluating the maximum error in the calculated surface area yields 26101.410.**

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