Question

the length of a rectangle is 5 more than the width. the area is 126 square feet. find the length and width of the rectangle.

Explanation:

Define variables and set up the equation

Let the width of the rectangle be \(w\) feet.
Since the length is \(5\) more than the width, the length is \(w + 5\) feet.
Using the Geometric Quadratic Modeling concept, the area of a rectangle is length times width:

$$\text{Area} = w(w + 5)$$

We are given that the area is \(126\) square feet, so we set up the equation:

$$w(w + 5) = 126$$

Write the quadratic equation in standard form

Using the Translating Word Problems to Quadratic Equations concept, we expand and rearrange the equation into standard form \(aw^2 + bw + c = 0\):

$$w^2 + 5w = 126$$

Subtract \(126\) from both sides:

$$w^2 + 5w - 126 = 0$$

Factor the quadratic equation

Using the Solving Quadratic Equations concept, we find two numbers that multiply to \(-126\) and add to \(5\).
These numbers are \(14\) and \(-9\).
Thus, we can factor the quadratic expression:

$$(w + 14)(w - 9) = 0$$

Solve for the width

Using the Zero Product Property, we set each factor to zero:

$$w + 14 = 0 \implies w = -14$$

$$w - 9 = 0 \implies w = 9$$

Since the width of a rectangle must be a positive value, we discard \(w = -14\).
Therefore, the width is \(w = 9\) feet.

Calculate the length

Using the relationship for length:

$$\text{Length} = w + 5 = 9 + 5 = 14\text{ feet}$$

Answer:

The length of a rectangle is 5 more than the width. The area is 126 square feet. Find the length and width of the rectangle.

Width = <blank>9</blank> feet

Length = <blank>14</blank> feet