Problem 60 - Algebra for College Students

Question

Set up an algebraic equation and solve each problem. The perimeter of a rectangle is 114 centimeters. If the ratio of its width to its length is 7 to 12 , find the dimensions of the rectangle.

Step-by-Step Solution

Answer

The rectangle's dimensions are 21 cm by 36 cm.

Step 1: Understand the problem

The problem involves finding the dimensions of a rectangle given its perimeter and the ratio of its width to its length. The perimeter of a rectangle is given as 114 centimeters, and the width-to-length ratio is 7:12.

Step 2: Define variables

Let the width of the rectangle be represented as \(7x\) and the length as \(12x\), where \(x\) is a common factor. These expressions reflect the given ratio of 7:12 for width to length.

Step 3: Write the perimeter formula

The perimeter (P) of a rectangle is given by the formula: \[ P = 2( ext{length} + ext{width}) \] Substituting the expressions for length and width, the equation becomes: \[ 114 = 2(12x + 7x) \]

Step 4: Simplify the equation

Simplify the expression within the parenthesis: \[ 12x + 7x = 19x \] Now, substitute back into the equation:\[ 114 = 2(19x) \]

Step 5: Solve for x

Divide both sides of the equation by 2 to isolate \(19x\): \[ 57 = 19x \] Then divide both sides by 19 to solve for \(x\): \[ x = 3 \]

Step 6: Calculate the dimensions

Multiply \(x\) by 7 to find the width and by 12 to find the length: \[ ext{Width} = 7x = 7(3) = 21 ext{ cm} \] \[ ext{Length} = 12x = 12(3) = 36 ext{ cm} \] Thus, the dimensions of the rectangle are 21 cm (width) and 36 cm (length).

Key Concepts

Perimeter of a Rectangle

Understanding the perimeter of a rectangle is fundamental to solving many geometric problems. The perimeter is simply the total distance around the rectangle. It is calculated by adding up the lengths of all sides of the rectangle. For a rectangle with sides of length "l" and width "w", the formula is:
\[ P = 2(l + w) \]
This formula reflects the fact that a rectangle has two pairs of equal sides. By multiplying the sum of the length and width by 2, we account for all four sides.

• If you know the perimeter and one side length, you can rearrange the formula to find the missing dimension.
• Understanding this formula helps in problems where dimensions are not directly given and need to be derived from other information, such as a given perimeter and a ratio between dimensions.

This concept is fundamental when dealing with algebraic equations that involve geometric shapes.

Ratio and Proportion

The concepts of ratio and proportion are essential in mathematics. They allow us to compare sizes and amounts, indicating how much of one quantity exists compared to another.
A ratio is a way to express a relationship between two quantities, showing how many times one value contains or is contained within the other. Here, the given ratio of the width to length of the rectangle is 7:12. This tells us that for every 7 units of width, there are 12 units of length.

• To use a ratio in solving problems, equate it to expressions involving variables, like in the rectangle's dimensions which were expressed as \(7x\) and \(12x\).
• By maintaining the ratio while solving equations, you ensure that the relationship holds true for the dimensions of the rectangle.

Proportion, on the other hand, is an equation that states two ratios are equivalent, a concept often used alongside ratios when solving equations.

Solving for Variables

In algebra, solving for a variable means finding the value of a variable that makes an equation true. This often involves manipulating equations to isolate the variable you are solving for.
Let's walk through the process using our rectangular problem:

• First, identify what you need to solve. For the rectangle with a perimeter of 114 cm and given ratio, you need the dimensions.
• Set up an equation using the perimeter and the ratio expressions: \[ 114 = 2(12x + 7x) \].
• Simplify and solve this equation by performing mathematical operations that keep the equation balanced. Here, simplifying gives \(114 = 38x\).
• The final step is isolating \(x\) by dividing, giving \(x = 3\).

This solution method highlights the importance of variable manipulation and balance in algebra, ensuring you find accurate values.