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Chapter 4: Problem 50

The perimeter of a rectangle is \(66 \mathrm{in}\). The length is twice as long as the width. Find the length and width.

Short Answer

Expert verified
Width: 11 in, Length: 22 in

Step by step solution

01

Understand the Perimeter Formula

The formula for the perimeter of a rectangle is given by: \[ P = 2(l + w) \] where \(P\) is the perimeter, \(l\) is the length, and \(w\) is the width.
02

Plug in the Given Perimeter

Given that the perimeter \(P\) is \(66\) inches, we can write: \[ 66 = 2(l + w) \]
03

Set Up the Relationship Between Length and Width

It is given that the length \(l\) is twice as long as the width \(w\). Thus, we can write: \[ l = 2w \]
04

Substitute the Relationship into the Perimeter Equation

Substitute \(l = 2w\) into the perimeter equation: \[ 66 = 2(2w + w) \]
05

Simplify and Solve for Width

Simplify the equation: \[ 66 = 2(3w) \] \[ 66 = 6w \] Divide both sides by \(6\): \[ w = \frac{66}{6} \] \[ w = 11 \]
06

Find the Length

Using the relationship \(l = 2w\), substitute \(w = 11\): \[ l = 2(11) \] \[ l = 22 \]

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Key Concepts

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

Perimeter Formula
Understanding the perimeter formula is crucial for solving rectangle-related problems. The perimeter of any polygon is the total distance around it.
For a rectangle, this can be specifically calculated using the formula:

\[ P = 2(l + w) \]

This signifies that the perimeter (P) is twice the sum of the rectangle's length (l) and width (w). The factor of 2 comes in because a rectangle has two pairs of opposite sides with equal length.
  • Imagine walking around the rectangle's border.
  • You trace two lengths and two widths.
  • The formula captures this repetition by multiplying by 2.

Once you grasp this fundamental concept, applying it to specific problems becomes more straightforward.
Linear Equations
Linear equations help in expressing relationships between variables using equalities. In the given problem, such a relationship is set up by the information that the length is twice the width. Mathematically, this:\[ l = 2w \]

This equation defines how one variable depends on another. When we have such information, it allows us to:
  • Substitute one variable in terms of another.
  • Simplify complex equations into manageable forms.

By substituting \ l = 2w \ into the perimeter equation, we get:

\[ 66 = 2(2w + w) \]

Here, we've replaced l with 2w, simplifying it to one variable. Such substitution techniques are common in solving linear equations. You solve for one variable to understand overall relationships in geometric problems.
Geometry
Geometry helps us understand the properties and relationships of shapes. A rectangle is a basic but essential geometric shape. Key properties include:
  • Opposite sides are equal and parallel.
  • It has four right angles.

    • Using these properties in the problem, we know:

    \[ l + w \]represents the sum of one pair of opposite sides.

    By understanding the constraints given (perimeter of 66 inches and length being twice the width), we translated it into a solvable equation.

    The solutions found were:\[ l = 22 \mathrm{in} \] and\[ w = 11 \mathrm{in} \]

    This shows the practical application of geometric principles and helps in visually validating our mathematical solutions.

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Most popular questions from this chapter

A glass contains 6 oz of cranberry juice drink. This drink is \(24 \%\) cranberry juice. Find the amount of pure cranberry juice in this glass.

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For exercises 29-34, a karat describes the percent gold in an alloy (a mixture of metals). $$ \begin{array}{|c|c|} \hline \text { Name of alloy } & \text { Percent gold } \\ \hline \text { 10-karat gold } & 41.7 \% \\ \text { 14-karat gold } & 58.3 \% \\ \text { 18-karat gold } & 75 \% \\ \text { 20-karat gold } & 83.3 \% \\ \text { 24-karat gold } & 100 \% \\ \hline \end{array} $$ Find the amount of 14-karat gold and the amount of 20 -karat gold to combine to make 8 oz of \(18-k a r a t\) gold. Round to the nearest hundredth.
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