To solve this problem, we need to find an expression for the width of the rectangle. Here’s a breakdown of the steps:

Given Information

• The area of the rectangle is $x^2 - 11x + 30$ square meters.
• The length of the rectangle is given as $x - 5$ meters.

Formula for the Area of a Rectangle

For a rectangle, the area is given by: $\text{Area} = \text{Length} \times \text{Width}$ Since we know both the area and the length, we can rearrange this formula to solve for the width: $\text{Width} = \frac{\text{Area}}{\text{Length}}$

Step 1: Substitute the Known Values

Substitute $x^2 - 11x + 30$ for the area and $x - 5$ for the length: $\text{Width} = \frac{x^2 - 11x + 30}{x - 5}$

Step 2: Simplify the Expression

To simplify, we can factor the quadratic expression in the numerator, $x^2 - 11x + 30$.

Looking for factors of 30 that add up to -11, we find: $x^2 - 11x + 30 = (x - 5)(x - 6)$ So, we rewrite the width as: $\text{Width} = \frac{(x - 5)(x - 6)}{x - 5}$

Step 3: Cancel Common Terms

Since $x - 5$ appears in both the numerator and the denominator, we can cancel it out (as long as $x \neq 5$): $\text{Width} = x - 6$

Answer

The expression that represents the width of the rectangle is: $\text{Width} = x - 6 \text{ meters}$

Would you like a detailed explanation of any step?

Here are some related questions to explore:

1. How can you factor other quadratic expressions like $x^2 - 11x + 30$?
2. What would the width be if $x = 10$?
3. How do you handle expressions that don’t factor as easily?
4. Can the same method apply if the length expression were more complex?
5. What other applications involve dividing polynomials?

Tip: Always check if you can factor a quadratic expression first; it often simplifies the problem significantly.