4. Applications of Derivatives
Differentials
Problem 4.6.59
Textbook Question
Approximating changes
Approximate the change in the lateral surface area (excluding the area of the base) of a right circular cone of fixed height h = 6m when its radius decreases from r = 10 m to r = 9.9 m (S = πr√(r² + h²).
Verified step by step guidance1
Identify the formula for the lateral surface area of a right circular cone, which is given by S = πr√(r² + h²).
Substitute the fixed height h = 6 m into the formula to express S in terms of r only: S = πr√(r² + 36).
Calculate the derivative of S with respect to r, S'(r), using the product and chain rules to find how S changes as r changes.
Evaluate S'(r) at the initial radius r = 10 m to find the rate of change of the lateral surface area at that point.
Multiply the rate of change S'(10) by the change in radius (Δr = 9.9 m - 10 m) to approximate the change in lateral surface area.
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