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Chapter 3: Problem 80

If the length of a rectangle is 4 more than 5 times the width and the perimeter is 32 meters, what are its dimensions?

Short Answer

Expert verified
The dimensions are 2 meters (width) and 14 meters (length).

Step by step solution

01

Identify Variables

Let the width of the rectangle be denoted as \( w \) meters. The length of the rectangle can be expressed in terms of the width as \( l = 5w + 4 \) meters.
02

Write the Perimeter Formula

The formula for the perimeter \( P \) of a rectangle is given by \( P = 2l + 2w \). Here, it is stated that the perimeter is 32 meters.
03

Express Perimeter in Terms of Width

Substitute the given values and the expression for the length (\( l = 5w + 4 \)) into the perimeter formula: \[ 32 = 2(5w + 4) + 2w \]
04

Simplify the Equation

Expand and simplify the equation: \[ 32 = 10w + 8 + 2w \] which reduces to \[ 32 = 12w + 8 \]
05

Solve for Width

Isolate \( w \) by subtracting 8 from both sides: \[ 32 - 8 = 12w \] which simplifies to \[ 24 = 12w \]. Then divide both sides by 12: \[ w = 2 \]
06

Find the Length

Substitute \( w = 2 \) back into the expression for length: \( l = 5w + 4 \), resulting in \( l = 5(2) + 4 = 14 \)
07

State the Dimensions

The width of the rectangle is 2 meters and the length is 14 meters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perimeter of a Rectangle
Understanding the perimeter of a rectangle is key to solving many geometry problems. The perimeter is the total distance around the rectangle. It can be found by adding up all the sides. Since a rectangle has opposite sides that are equal in length, we use the following formula: \[ P = 2l + 2w \] This means you add the length (\

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