Approximate the change in the lateral surface area (excluding the area of the base) of a right circular cone of fixed height of hequals=88 m when its radius decreases from requals=99 m to requals=8.98.9 m (upper s equals pi r startroot r squared plus h squared endroots=πrr2+h2).

S = πr√(r^2+h^2)

h = 8 m (constant)

An approximation of S when r changes from 9 to 8.9

Such an approximation is usually made by estimating the change using the first derivative. That derivative with respect to r is

... S' = π√(r^2+h^2) + πr(1/2·r)/√(r^2+h^2)

... S' = π(2r^2 +h^2)/√(r^2 +h^2) . . . . . use a common denominator

For r=9, h=8, this is

... S' = π(2·81 +64)/√(81+64) = 226π/√145 ≈ 58.96

Then the change in lateral surface area will be approximately

... ∆S ≈ (∆r)·S' ≈ (-0.1)·(58.96) ≈ -5.90 . . . m²

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