Question

The length of a rectangular field is 20 less than its width. The area of the field is 12,000 ft2. What is the width of the field

Answer 1

The answer is 120 feet.

The area of the field (A) is:
A = w · l       (w - width, l - length)

It is known:
A = 12,000 ft²
l = w - 20
So, let's replace this in the formula for the area of the field:
12,000 = w · (w - 20)
12,000 = w² - 20
⇒ w² - 20w - 12,000 = 0

This is quadratic equation. Based on the quadratic formula:
ax² + bx + c = 0      ⇒ $x= \frac{-b+/- \sqrt{ b^{2}-4ac } }{2a }$

In the equation w² - 20w - 12,000 = 0, a = 1, b = -20, c = -12000
Thus:
$w= \frac{-(-20)+/- \sqrt{(-20)^{2}-4*1*(-12000) } }{2*1} = \frac{20+/- \sqrt{400+48000} }{2} = \frac{20+/-220}{2}$
So, width w can be either
$w= \frac{20+220}{2}= \frac{240}{2}=120$
or
$w= \frac{20-220}{2}= \frac{-200}{2} =-100$
Since, the width cannot be a negative number, the width of the field is 120 feet.