Linear Approximations: Quiz
Problem 5
A lawnmower using a 40-foot extension cord cuts grass in a circular area. It is discovered that there is a knot in the cord, and untying it lengthens the cord to 42 feet. Using an appropriate linearization, estimate how much more area may be cut with the unknotted cord.
Answers, problem 5
The formula for area of the circle is A = πr2. A linearization of this function about r = 40 is L(x) = π(40)2 + 80π(x − 40). An estimate of the new area is L(42) = π(40)2 + 80π(2) = 1760π, so the additional area is 160π square feet.