find the area of the rectangle perimeter is 130 length is (3x-15) width (x-8) find the area

To find the area of the rectangle, we need to first determine the values of the length and width using the given information, and then calculate the area.

### Step 1: Set up the equation for perimeter
The perimeter \(P\) of a rectangle is given by the formula:

\[
P = 2(\text{Length} + \text{Width})
\]

We are told that the perimeter is 130, the length is \(3x - 15\), and the width is \(x - 8\). Substituting these values into the formula:

\[
130 = 2\left((3x - 15) + (x - 8)\right)
\]

### Step 2: Solve for \(x\)
Simplify the equation:

\[
130 = 2\left(4x - 23\right)
\]

Divide both sides by 2:

\[
65 = 4x - 23
\]

Add 23 to both sides:

\[
88 = 4x
\]

Now, divide by 4:

\[
x = 22
\]

### Step 3: Find the length and width
Now that we have \(x = 22\), we can substitute this value back into the expressions for the length and width:

\[
\text{Length} = 3x - 15 = 3(22) - 15 = 66 - 15 = 51
\]

\[
\text{Width} = x - 8 = 22 - 8 = 14
\]

### Step 4: Calculate the area
The area \(A\) of a rectangle is given by the formula:

\[
A = \text{Length} \times \text{Width}
\]

Substitute the values for the length and width:

\[
A = 51 \times 14 = 714
\]

So, the area of the rectangle is \(714 \, \text{square units}\).

Would you like further details or clarification? Here are some related questions:

1. What if the perimeter was different? How would it change the area?
2. How does changing \(x\) affect the length and width of the rectangle?
3. What is the effect on the perimeter when length or width is doubled?
4. Can we express the area as a function of \(x\) before solving for \(x\)?
5. How would the problem change if we knew the area but not the perimeter?

**Tip:** Always check if you need to solve for variables first when working with formulas.

Learn how to find the area of a rectangle when given its perimeter of 130 units, length as (3x-15), and width as (x-8). Follow the step-by-step algebraic solution to solve for x and then calculate the area. Ideal for students in Grades 7-9, this problem covers key geometry and algebra concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation