Problem 48 - 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. Let \(h\) represent the average height of the other stories. Finish assigning labels and write the algebraic model.

Step-by-Step Solution

Answer

The average height of the other stories (h) in the building is 12 feet.

Step 1: Identifying the Known and Unknown

We are given the total height of the first 7 stories of the building to be 126 feet. We also know that if these 7 stories were of average height, then the building could have fit about 3.5 more stories, which is equivalent to saying the total height could contain 10.5 stories. Let \(h\) represent the average height of the other stories.

Step 2: Setting Up the Equations

With \(h\) representing the average height of a story in the remaining 43 floors, the equation becomes: 126 = 10.5h. The equation is set up as such because if the first 7 stories were of average height, we could have fit an additional 3.5 stories, making the total number of stories 10.5. Therefore, the total height of these 10.5 stories at average height(h) would still be equal to the actual combined height of the first seven stories (which is 126 feet).

Step 3: Solving for h

To solve for \(h\), we simply need to divide 126 by 10.5. Doing this gives us \(h = 12\) feet. This tells us that the average height of the remaining stories in the building is 12 feet.

Key Concepts

Algebraic Equations

Algebraic equations are the backbone of algebraic modeling, serving as critical tools for translating real-world problems into mathematical expressions. An algebraic equation is a statement of equality that involves variables and constants. The equation typically includes operators such as addition, subtraction, multiplication, and division, and it can be as simple as a linear equation or as complex as a quadratic equation or higher polynomials.

For example, if we know the combined height of the first seven stories of a building and the additional number of stories it could accommodate at an average height, we can set up an equation to find this unknown average height. In the given exercise, the equation is constructed as follows: the combined height of 126 feet is equal to 10.5 times the average height of a story, written as the algebraic equation \(126 = 10.5h\). Here, \(h\) is the variable representing the average height of a story, and by solving the equation, we can determine its value.

Problem-solving in Algebra

The process of problem-solving in algebra involves several systematic steps that move from understanding the given information to arriving at a solution. In the context of algebraic modeling, these steps often include identifying what we know and what we need to find out (variables), formulating equations, and then solving these equations.

In the building height problem, we first identified the known heights and the conceptual relationship between the total height and the number of stories. We then defined the variable \(h\) and formulated the equation based on the given information. Solving the equation involved dividing the total known height by the hypothetical number of stories if all were of average height (126 / 10.5). The steps taken provide not only the solution—finding that the average height is 12 feet—but also illustrate a clear path to understanding how the components of the problem relate to each other.

Real-world Applications of Algebra

Algebra finds its practicality in numerous real-world applications, from architecture and engineering to economics and health sciences. It helps in creating models that can predict, analyze, and solve tangible problems. The exercise involving the Grand Gateway at Xu Hui in Shanghai is a real-world application where algebra aids in building design.

Understanding how many stories can fit in a specific vertical space or determining the potential for additional construction requires algebraic modeling. The equation \(126 = 10.5h\) derived from the building's story height translates into practical insights for architects and engineers. By manipulating this equation, stakeholders can make informed decisions about building design, cost estimation, or even zoning regulations. As such, algebraic modeling not only supports theoretical learning but also empowers professionals to make data-driven decisions in the physical world.