Problem 17 - Algebra 1: Concepts and Skills

Question

Solve the formula for the indicated variable. Show all your steps. Then evaluate the new formula by substituting the given values. The formula for the perimeter of a rectangle is \(P=2 \ell+2 w .\) Find the area of a rectangle that has perimeter \(P\) of 18 centimeters and length \(\ell\) of 6 centimeters. HINT: You can begin by solving the perimeter formula for \(w\).

Step-by-Step Solution

Answer

The area of the rectangle is 18 square centimeters.

Step 1: Solve for \(w\)

Firstly, take the given perimeter formula \(P = 2l + 2w\). Then, solve for \(w\) by subtracting \(2l\) from both sides, giving you: \(w = (P - 2l) / 2\).

Step 2: Substitute Given Values

Next, substitute \(P = 18\) and \(l = 6\) into the formula for \(w\). That will give you: \(w = (18 - 2*6) / 2 = 3\).

Step 3: Calculate Area

Finally, find the area of the rectangle by multiplying the length (\(l = 6\)) by the calculated width (\(w = 3\)). This gives you: \(Area = l * w = 6 * 3 = 18\) square centimeters.

Key Concepts

Solving Equations

Solving an equation involves finding the value of a variable that makes the equation true. In this exercise, the equation we start with is the perimeter formula for a rectangle, which is \( P = 2\ell + 2w \). Here, \( P \) stands for the perimeter, \( \ell \) for the length, and \( w \) for the width.
To solve this equation for \( w \), follow these steps:

• First, isolate \( 2w \) by subtracting \( 2\ell \) from both sides of the equation. This gives \( 2w = P - 2\ell \).
• Then, divide everything by 2 to solve for \( w \), resulting in \( w = (P - 2\ell) / 2 \).

By rearranging the equation, we can find the value of the width when we know the perimeter and the length.

Area of Rectangle

The area of a rectangle is a measure of the amount of space inside the rectangle. It's calculated using the formula \( Area = \ell \times w \), where \( \ell \) is the length and \( w \) is the width of the rectangle.
Once we've calculated the width \( w \) using the perimeter formula and the given perimeter value, we can easily find the area. Given our rectangle's length of 6 centimeters and width of 3 centimeters as solved before, the area is computed as follows:

• Substitute the length \( \ell = 6 \) and width \( w = 3 \) into the area formula.
• Calculate \( Area = \ell \times w = 6 \times 3 = 18 \) square centimeters.

This calculation shows that the rectangle covers an area of 18 square centimeters.

Geometry Formulas

In geometry, formulas are essential tools that enable us to calculate measurements like area, perimeter, and volume. For rectangles, two key formulas are commonly used:

• Perimeter Formula: \( P = 2\ell + 2w \). This formula calculates the total distance around the rectangle. It is derived from adding together the lengths of all four sides of the rectangle.
• Area Formula: \( Area = \ell \times w \). This formula finds the total space inside the rectangle by multiplying the length by the width.

Understanding these formulas is crucial for solving geometry problems that involve rectangles. By mastering these concepts, you can solve real-world problems like designing a garden or finding the space inside a room.