Question

1. one saturday morning george goes out to cut his lot that is 100 ft by 120 ft. he starts cutting around the outside boundary spiraling around towards the center. by noon he has cut 60% of the lawn. what is the width of the ring that he has cut?

Explanation:

Step1: Calculate total lawn area

The area of the rectangular lawn $A = 100\times120=12000$ square - feet.

Step2: Calculate the area cut

The area cut is $60\%$ of the total area, so $A_{cut}=0.6\times12000 = 7200$ square - feet.

Step3: Let the width of the ring be $x$ feet

The un - cut area forms a smaller rectangle with length $l=(120 - 2x)$ feet and width $w=(100 - 2x)$ feet. The area of the un - cut rectangle $A_{un - cut}=(120 - 2x)(100 - 2x)$.

Step4: Set up an equation

Since $A_{cut}=A - A_{un - cut}$, we have $7200=12000-(120 - 2x)(100 - 2x)$.
Expand the right - hand side:
$(120 - 2x)(100 - 2x)=120\times100-120\times2x-2x\times100 + 4x^{2}=12000-440x + 4x^{2}$.
The equation becomes $7200=12000-(12000-440x + 4x^{2})$.
Simplify the equation:
$7200=440x - 4x^{2}$.
Rearrange to get a quadratic equation: $4x^{2}-440x + 7200 = 0$.
Divide through by 4: $x^{2}-110x + 1800 = 0$.

Step5: Solve the quadratic equation

For a quadratic equation $ax^{2}+bx + c = 0$ ($a = 1$, $b=-110$, $c = 1800$), the quadratic formula is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$.
First, calculate the discriminant $\Delta=b^{2}-4ac=(-110)^{2}-4\times1\times1800=12100 - 7200 = 4900$.
Then $x=\frac{110\pm\sqrt{4900}}{2}=\frac{110\pm70}{2}$.
We have two solutions:
$x_1=\frac{110 + 70}{2}=\frac{180}{2}=90$ (rejected since $120-2x\gt0$ and $100 - 2x\gt0$, if $x = 90$, $120-2x=120 - 180=-60\lt0$ and $100 - 2x=100 - 180=-80\lt0$).
$x_2=\frac{110 - 70}{2}=\frac{40}{2}=20$.

Answer:

$20$ feet