A lawnmower using a 40 m 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 m. Using an appropriate linearization, estimate how much more area can be cut with the unknotted cord. A) 160 $\pi$m² B) 80 $\pi$ m² C) 120 $\pi$ m² D) 320 $\pi$ m² E) 225 $\pi$ m²

Question

Quizplus · Accepted Answer

The area covered by the lawnmower with a cord of length $L$ is given by the formula for the area of a circle, $A = \pi r^2$, where $r$ is the radius of the circle. Since the length of the cord is equal to the radius, we can write $A = \pi L^2$. The difference in area ($\Delta A$) due to an increase in the length of the cord ($\Delta L$) can be estimated using the derivative of $A$ with respect to $L$, which gives us $dA/dL = 2\pi L$. For a small change in $L$, $\Delta A \approx dA/dL \cdot \Delta L = 2\pi L \cdot \Delta L$. With $L = 40$ m and $\Delta L = 2$ m, we get $\Delta A \approx 2\pi \cdot 40 \cdot 2 = 160\pi$ m$^2$.

A Lawnmower Using a 40 M Extension Cord Cuts Grass π\piπ

A Lawnmower Using a 40 M Extension Cord Cuts Grass $\pi$