Problem 113 - Elementary Algebra

Question

The perimeter of a rectangle is \(84 .\) The length is 10 more than three times the width. Find the length and width.

Step-by-Step Solution

Answer

The width is 8 units and the length is 34 units.

Step 1: Define Variables

Let the width of the rectangle be denoted by \(w\) and the length be denoted by \(l\).

Step 2: Set Up Relationships

According to the problem, the length \(l\) is 10 more than three times the width. So, \(l = 3w + 10\).

Step 3: Use the Perimeter Formula

The perimeter \(P\) of a rectangle is given by the formula: \[P = 2l + 2w\]. We know the perimeter \(P = 84\).

Step 4: Substitute the Relationship into the Perimeter Formula

Substitute \(l = 3w + 10\) into the perimeter formula: \[84 = 2(3w + 10) + 2w\].

Step 5: Simplify and Solve for \(w\)

Distribute and combine like terms: \[84 = 6w + 20 + 2w\] \[84 = 8w + 20\]. Subtract 20 from both sides: \[64 = 8w\]. Divide by 8: \[w = 8\].

Step 6: Find the Length \(l\)

Use the width to find the length: \[l = 3w + 10\] \[l = 3(8) + 10\] \[l = 24 + 10\] \[l = 34\].

Key Concepts

rectangle perimeter

The perimeter of a rectangle is the total distance around the edge of the rectangle. In simpler terms, think of it as fences around a garden.
To find the perimeter, we add up all four sides. Since opposite sides of a rectangle are equal, we can use the formula: \[ P = 2l + 2w \] where \( l \) is the length and \( w \) is the width.
In this problem, the perimeter is given as 84. Using the formula, we set up our equation as \[ P = 2l + 2w = 84 \].
Knowing the perimeter helps us form an equation, which will be crucial in solving for missing dimensions.

variable substitution

Variable substitution is a key technique in solving equations, especially in word problems. It involves replacing one variable with an expression involving another variable.
In our problem, we are told that the length \( l \) is 10 more than three times the width \( w \). This relationship can be written as the equation: \[ l = 3w + 10 \].
By substituting this expression into the perimeter formula, we can solve the equation with one variable.
Substituting \( l = 3w + 10 \) in the perimeter formula \[ P = 2l + 2w = 84 \]: \[ 84 = 2(3w + 10) + 2w \].
This transformation helps us handle equations that might look complicated at first.

solving linear equations

Solving linear equations involves finding the value of the variable that makes the equation true. These equations are usually in the form \( ax + b = c \).
In our substituted equation \[ 84 = 2(3w + 10) + 2w \], we start by simplifying: \[ 84 = 6w + 20 + 2w \].
Then, combine like terms: \[ 84 = 8w + 20 \].
We isolate the variable \( w \) by performing inverse operations. Subtract 20 from both sides: \[ 64 = 8w \].
Finally, divide by 8: \[ w = 8 \].
With \( w = 8 \), we use the original relationship \( l = 3w + 10 \) to find the length: \[ l = 3(8) + 10 \] \[ l = 24 + 10 \] \[ l = 34 \].
Thus, solving linear equations step-by-step ensures we find the correct dimensions.