Question

The length of a rectangle is 5 centimeters more than the width. The perimeter of the rectangle is 90 centimeters. What is the length of the rectangle? A. 15 centimeters B. 18 centimeters C. 22.5 centimeters D. 25 centimeters

Short answer

D. 25 centimeters

Step-by-step solution

Define Variables

Let the width of the rectangle be \( w \) centimeters. Then the length of the rectangle is \( w + 5 \) centimeters.

Perimeter Formula

Recall the formula for the perimeter of a rectangle: \[ P = 2(l + w) \] where \( l \) and \( w \) are the length and width respectively.

Set Up the Equation

Substitute the expressions for the perimeter and the length: \[ 90 = 2((w + 5) + w) \]

Simplify the Equation

Simplify inside the parentheses: \[ 90 = 2(2w + 5) \]

Solve for Width

Divide both sides of the equation by 2: \[ 45 = 2w + 5 \] \Subtract 5 from both sides: \[ 40 = 2w \]Divide by 2: \[ w = 20 \]

Find the Length

Since the length is 5 centimeters more than the width: \[ l = w + 5 = 20 + 5 = 25 \] So the length of the rectangle is 25 centimeters.

Key concepts

length of rectangle

In this problem, we need to find the length of a rectangle. The information given is that the length is 5 centimeters more than the width. You can express this relationship using a simple algebraic expression. If we let the width be denoted by the variable \( w \), then the length becomes \( w + 5 \). This idea is essential because it allows us to use algebra to find the missing dimensions of a rectangle.

perimeter formula

To solve this type of geometry problem, one crucial formula to know is the perimeter formula of a rectangle. The perimeter \( P \) of a rectangle is given by the equation: \[ P = 2(l + w) \] where \( l \) is the length, and \( w \) is the width. Here, we're told that the perimeter is 90 centimeters. By substituting the values, we can form an equation to solve.

algebraic equations

Algebra is fundamental in solving geometry problems like this. Here’s how we set up and solve the equation. We start by plugging in the length \( l = w + 5 \, \) and the perimeter: \[ 90 = 2((w + 5) + w) \] Next, simplify inside the parentheses: \[ 90 = 2(2w + 5) \] Then, isolate the variable. Divide by 2: \[ 45 = 2w + 5 \] Subtract 5: \[ 40 = 2w \] Finally, divide by 2 to find the width: \[ w = 20 \]

rectangular dimensions

Once we have the width, finding the length is straightforward. Since the length is 5 centimeters more than the width: \[ l = w + 5 \left( 20 + 5 = 25 \right) \] Thus, the length of the rectangle is 25 centimeters. Now, you know how to determine the dimensions of a rectangle by understanding the relationship between the sides, using perimeter formulas, and solving algebraic equations.