3. The area of a rectangle is $$ x^{2}-x-2 $$ and the length of the rectangle is $$ x+1 $$. a. Find the width of the rectangle. b. Find the area of the rectangle if the width is 4 m .

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UpStudy AI · Accepted Answer

**Step 1. Express the area in terms of length and width**

The area $$ A $$ of a rectangle is given by:
$$
A = \text{length} \times \text{width}
$$
Given that:
$$
A = x^2 - x - 2 \quad \text{and} \quad \text{length} = x + 1,
$$
we can set up the equation:
$$
x^2 - x - 2 = (x + 1) \times \text{width}.
$$

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**Step 2. Find the width of the rectangle (Part a)**

To find the width, solve for the width:
$$
\text{width} = \frac{x^2 - x - 2}{x + 1}.
$$
Factor the quadratic in the numerator:
$$
x^2 - x - 2 = (x + 1)(x - 2).
$$
Then:
$$
\text{width} = \frac{(x + 1)(x - 2)}{x + 1}.
$$
Assuming $$ x + 1 \neq 0 $$ (i.e., $$ x \neq -1 $$), cancel $$ x + 1 $$:
$$
\text{width} = x - 2.
$$

---

**Step 3. Find the area when the width is 4 m (Part b)**

Given the width:
$$
x - 2 = 4.
$$
Solve for $$ x $$:
$$
x = 4 + 2 = 6.
$$
Now substitute $$ x = 6 $$ into the area formula:
$$
A = x^2 - x - 2 = 6^2 - 6 - 2.
$$
Calculate:
$$
A = 36 - 6 - 2 = 28.
$$
Thus, the area is $$ 28 $$ square meters.

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**Final Answers:**

a. The width of the rectangle is $$ x - 2 $$.

b. If the width is 4 m, the area of the rectangle is $$ 28 $$ m$$^2$$.
a. The width of the rectangle is $$ x - 2 $$. b. If the width is 4 m, the area of the rectangle is 28 m².

The area of a rectangle is and the length of the rectangle is .
a. Find the width of the rectangle.
b. Find the area of the rectangle if the width is 4 m .

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The area of a rectangle is and the length of the rectangle is . a. Find the width of the rectangle. b. Find the area of the rectangle if the width is 4 m .