Problem 24 - Intermediate Algebra

Question

New York City. The triangular-shaped Flatiron Building in Manhattan has a perimeter of 499 feet at its base. It is bordered on each side by a street. The 5 th Avenue front of the building is 198 feet long. The Broadway front is 43 feet more than twice as long as the East 22nd Street front. Find the length of the Broadway front and East 22nd Street front. (Source: New York Public Library)

Step-by-Step Solution

Answer

East 22nd Street is 86 feet, and Broadway is 215 feet long.

Step 1: Understand the Problem

We need to find the lengths of two sides of a triangular building's base: Broadway and East 22nd Street fronts. We know the total perimeter is 499 feet, the 5th Avenue front is 198 feet, and Broadway is 43 feet longer than twice the East 22nd Street front.

Step 2: Define Variables

Let the length of the East 22nd Street front be \( x \) feet. The length of the Broadway front is then \( 2x + 43 \) feet.

Step 3: Set Up the Perimeter Equation

The total perimeter of the building at the base can be expressed as the sum of the lengths of the three sides:\[ 198 + x + (2x + 43) = 499 \]

Step 4: Simplify the Equation

Combine like terms in the equation:\[ 198 + x + 2x + 43 = 499 \]\[ 3x + 241 = 499 \]

Step 5: Solve for \( x \)

Subtract 241 from both sides of the equation to isolate the terms with \( x \):\[ 3x = 258 \]Then divide by 3:\[ x = 86 \]

Step 6: Find the Broadway Length

Substitute \( x = 86 \) back into the equation for the length of the Broadway front:\[ 2x + 43 = 2(86) + 43 = 172 + 43 = 215 \]

Step 7: State the Lengths of the Fronts

The East 22nd Street front is 86 feet long, and the Broadway front is 215 feet long.

Key Concepts

Triangular Perimeter Problem

The triangular perimeter problem can be a fun challenge because it combines shapes and algebra. When dealing with a triangle, like the base of Manhattan's Flatiron Building, understanding perimeter is key.
The perimeter of a triangle is simply the combined length of its three sides. If you know the length of two sides, you can find the third side using the perimeter formula, and vice versa.
In our problem, we're interested in a triangle with a known total perimeter of 499 feet. One side, the 5th Avenue front, measures 198 feet. The task is to determine how long the Broadway and East 22nd Street fronts are, using algebra and the given relationships.
Think about how each side's length contributes to the total perimeter, and this will guide you to setting up equations for solutions. It involves a step-by-step break down starting from understanding known values, to using equations to find the unknowns.

System of Equations

When working on algebra word problems like our triangular perimeter problem, formulating a system of equations is a powerful tool. In essence, a system of equations consists of two or more equations using the same variables. Solving these simultaneously helps us find values for those variables.
In this problem, we designate the East 22nd Street front as variable \( x \), and represent the Broadway front based on \( x \) as \( 2x + 43 \), given Broadway's relationship to the East 22nd Street front.
This approach helps break down what seems like a complex situation into manageable mathematical tasks:

• First, writing the perimeter equation: \( 198 + x + (2x + 43) = 499 \)
• Combining like terms to simplify this lineup of numbers.

The system of equations approach helps visualize and work through relationships between the sides, offering a clear course to determine unknown lengths.

Variable Substitution

Variable substitution is an essential concept in algebra that aids in solving equations efficiently. This technique involves replacing a variable with a numerical value or an equivalent expression that represents it.
Let’s look into how this works in our specific triangular perimeter problem. We've assigned \( x \) as the length of the East 22nd Street front.
The Broadway front is described as \( 2x + 43 \), which directly uses substitution to define its length in terms of \( x \).
Once you solve for \( x \), you perform substitution: plug \( x = 86 \) back into the expression to find the Broadway side’s specific length. Here's the substitution step: \( 2(86) + 43 = 215 \).
This skill of swapping a variable with its corresponding numeric solution allows you to solve for unknowns in a clear sequence, ensuring all parts of the solution fit together logically.