Problem 49 - Algebra 1

Question

Use the following information. In designing a tall building, many factors affect the height of each story. How the space will be used is important. At the Grand Gateway at Xu Hui in Shanghai, the lowest 7 stories have a combined height of about 126 feet. These stories next to a shopping mall are unusually tall. The building's other 43 stories have a more typical height. If the average height of the other stories had been used for the lowest 7 stories, then the building could have fit about \(3 \frac{1}{2}\) more stories. Solve the equation. Be sure to simplify the answer. About how many feet tall are the stories above the 7 th floor?

Step-by-Step Solution

Answer

The stories above the 7th floor are approximately 12 feet tall.

Step 1: Analyze and Understand the Problem

It's given that the first 7 floors have total height of 126 feet. If these seven floors had used the average height of the remaining floors, it would fit an additional \(3 \frac{1}{2}\) stories. This suggests that the average height of these seven floors is greater than the average height of the remaining floors. Let's denote the average height of the remaining stories, the one we want to find, as 'x' feet.

Step 2: Setup the Equation

The exercise states that \(3 \frac{1}{2}\) more stories could fit into the height of the first 7 floors if they had used the average height of the remaining floors. This means that the combined height of the first 7 floors, 126 feet, is equal to the height of \(7 + 3 \frac{1}{2}\) = \(10 \frac{1}{2}\) stories using the average height 'x'. Therefore, the equation becomes \(x * 10.5 = 126\).

Step 3: Solve the Equation

To find the average height 'x', divide both sides of the equation by 10.5. The solution is \(x = \frac{126}{10.5}\) feet.

Step 4: Simplify the Answer

Simplifying the fraction gives us that a single floor above the 7th floor is approximately 12 feet tall.

Key Concepts

Systems of Equations

Systems of equations are a collection of two or more equations with a set of variables. They are solved to find the value of the unknowns, and they are essential in modeling real-world problems. In this exercise, although we're not directly using multiple equations, the concept of equivalency helps in setting up the necessary equation.
By understanding how different variables interact with each other, we establish relationships like:

• Height of the first 7 stories as a total (126 feet).
• Possibility of adding more floors using average story height (10.5 stories).

These relationships hint at an implicit system where the solution involves finding common unknowns, like average story height (denoted as 'x'). Instead of adding several equations, we rely on one primary equation coming from understanding proportional changes. A systematic approach is vital, as seen when defining the unknown 'x' and how its value impacts the overall number of stories.

Average Height Calculation

Calculating averages is a fundamental mathematical skill. It captures a central value or "normal" measure among a set. In this case, you're determining the average story height of the building above the 7th floor.
We know:

• The combined height for the first 7 floors: 126 feet.
• Potential to fit these floors as 10.5 based on the new average height.

To find the average, you set up the equation for the height of 10.5 stories to equal the height of 7 stories: \(x \times 10.5 = 126\). Solving this requires dividing both sides by 10.5: \(x = \frac{126}{10.5}\).
Thus, each floor is about 12 feet tall after computation. The process here emphasizes converting total height into per-unit (per story) values, highlighting the concept of dividing totals by units.

Understanding and Analyzing Problems

Understanding and analyzing a problem involves deconstructing it into manageable parts. This exercise requires you to tease out underlying assumptions and premise each step properly.
Begin by identifying:

• The specific parameters given (first 7 floors total height, extra stories possible).
• The unknowns you need to solve for (average height of floors above).

The critical step is forming an equation that accurately represents the problem's conditions. Here, recognizing what these conditions mean in equation form (i.e., number of stories and average height) is vital. Analyzing allows you to spot relationships and dependencies between elements, ensuring the equation considers all parts of the problem. Through this analysis, you develop both the strategy (identifying 'x') and the calculation needed to find a solution, underscoring the strength of mathematical reasoning and problem-solving.