The floor of a shed has an area of 104 square feet. The floor is in the shape of a rectangle whose length is 3 feet less than twice the width. We need to find the length and the width of the floor of the shed.

1. Problem Description

The floor of a shed has an area of 104 square feet. The floor is in the shape of a rectangle whose length is 3 feet less than twice the width. We need to find the length and the width of the floor of the shed.

2. Solution Steps

Let $w$ be the width of the floor of the shed.

Let $l$ be the length of the floor of the shed.

The length is 3 feet less than twice the width, so we can write

$l = 2w - 3$.

The area of the rectangular floor is given by

Area = length * width, so $A = l * w$.

We are given that the area is 104 square feet, so $104 = l * w$.

Substituting $l = 2w - 3$ into the area equation, we get

$104 = (2w - 3)w$.

Expanding the equation gives

$104 = 2w^2 - 3w$.

Rearranging the equation into a quadratic form, we have

$2w^2 - 3w - 104 = 0$.

We can solve this quadratic equation for $w$ using the quadratic formula:

$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 2$, $b = -3$, and $c = -104$.

$w = \frac{3 \pm \sqrt{(-3)^2 - 4(2)(-104)}}{2(2)}$

$w = \frac{3 \pm \sqrt{9 + 832}}{4}$

$w = \frac{3 \pm \sqrt{841}}{4}$

$w = \frac{3 \pm 29}{4}$

We have two possible solutions for $w$:

$w = \frac{3 + 29}{4} = \frac{32}{4} = 8$

$w = \frac{3 - 29}{4} = \frac{-26}{4} = -6.5$

Since the width cannot be negative, we have $w = 8$.

Now we can find the length using $l = 2w - 3$:

$l = 2(8) - 3 = 16 - 3 = 13$.

So the width is 8 feet and the length is 13 feet.

We can check our answer: $8 * 13 = 104$, which is the given area.

3. Final Answer

The width of the floor of the shed is 8 ft.

The length of the floor of the shed is 13 ft.