Question

A rectangular lawn, 100 feet long by 50 feet wide, is to have two sidewalks installed that will cut the lawn in half both ways, the short way and the long way, meeting in the middle so that the lawn is divided into four equal sections. The sidewalk must occupy an area no more than 10% of the total lawn area. How wide can the sidewalk be?

Answer 1

Answer:

3.41 feet

Step-by-step explanation:

Area = Length × Breath

Area of the rectangular lawn = 100 × 50

                                                = 5000 feet²

The sidewalk must occupy an area no more than 10% of the total lawn area.

So, the area of the sidewalk would be not more than  = 10% × 5000

                                                                                         = 0.10 × 5000

                                                                                        = 500 feet²

Let the width of the sidewalk = x feet

area of the side walk = (L×W of the long way) + ((L-x)×W of the short way)

(100 × x) + ((50 - x) × x) < 500

100x + (50-x)(x) < 500

-x²  + 150x < 500

-x² + 150x = 500

-x² + 150x - 500 = 0

By using quadratic formula

$x=\frac{-b\pm\sqrt{b^2-4ac} }{2a}$

$x=\frac{(-150)\pm\sqrt{(150)^2-4(-1)(-500)} }{2(-1)}$

$x=\frac{-150\pm \sqrt{20500} }{-2}$

$x=75-5\sqrt{205}$ or $x=75+5\sqrt{205}$

x = 3.41089 ≈ 3.41 feet  or  x = 146.58

Therefore, width of the sidewalk would be 3.41 feet.