Question

A man mows his 200 ft by 40 ft rectangular lawn in a spiral pattern starting from the outside edge. After abit of hard work he stops for a water break, he is 78.75% done. How wide of a strip has he mowed aroundthe outside edge?

Answer 1

The above diagram represents the situtation.

The area of the whole lawn is:

200x40 = 8000 sq ft

If 78.75% of the lawn was mowed, then 21.25% ( = 100% - 78.75%) was not mowed. This area is:

8000x21.25% = 1700 sq ft

With the help of the diagram, this area is computed as follows:

`\begin{gathered} (40-2x)(200-2x)=1700 \\ 40\cdot200-40\cdot2x-2x\cdot200+2x\cdot2x=1700 \\ 8000-80x-400x+4x^2=1700 \\ 4x^2-480x+8000-1700=0 \\ 4x^2-480x+6300=0 \end{gathered}`

We can solve this equation with the help of the quadratic formula, as follows:

`\begin{gathered} x_(1,2)=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x_(1,2)=\frac{480\pm\sqrt[]{(-480)^2-4\cdot4\cdot6300}}{2\cdot4} \\ x_(1,2)=\frac{480\pm\sqrt[]{129600}}{8} \\ x_1=(480+360)/(8)=105 \\ x_2=(480-360)/(8)=15 \end{gathered}`

The first solution, x = 105 ft, has no sense in the context of the problem. It would be longer than the width of the rectangular lawn. In consequence, the correct answer is x = 15 ft.

The width of the mowed strip is 15 ft