Conference Table Width Calculation
In the realm of office space planning, the dimensions of a conference table play a pivotal role in fostering effective communication and collaboration. When designing or redesigning an office, one crucial aspect to consider is the size and shape of the conference table. This decision directly impacts the room's functionality and aesthetic appeal. In Mr. Nathan's office, the conference table's area is a key consideration, with a maximum limit set at 175 square feet. This constraint ensures that the table fits comfortably within the room without overwhelming the space or hindering movement.
Understanding the Constraints
The stipulation that the table's area must not exceed 175 square feet introduces a practical limitation that guides the selection of appropriate dimensions. This constraint prevents the selection of a table that is too large, which could impede workflow and make the room feel cramped. On the other hand, it also encourages the selection of a table that is sufficiently large to accommodate the typical number of attendees at meetings and discussions.
The Relationship Between Length and Width
The problem introduces an additional condition: the length of the table is 18 feet more than its width, denoted by x. This relationship establishes a direct link between the two dimensions, meaning that the length is dependent on the width. Mathematically, this can be expressed as length = x + 18. This equation forms the basis for determining the possible widths of the table while adhering to the area constraint.
Setting Up the Inequality
The area of a rectangular table is calculated by multiplying its length and width. Given the condition that the area must be no more than 175 square feet, we can set up an inequality to represent this constraint. The inequality is expressed as x(x + 18) ≤ 175, where x represents the width of the table. This inequality captures the essence of the problem, allowing us to mathematically explore the feasible range of widths. Solving this inequality will provide the interval of possible widths that satisfy the given conditions.
Importance of Practical Considerations
While mathematical solutions provide a range of possible widths, practical considerations also play a significant role in determining the optimal width. Factors such as the number of regular attendees at meetings, the need for space to spread out documents or equipment, and the overall layout of the room all influence the ideal table size. A table that is too small may feel cramped and limit productivity, while a table that is too large may waste valuable space and hinder communication.
Conclusion
The problem of determining the possible widths of the conference table in Mr. Nathan's office highlights the interplay between mathematical constraints and practical considerations in office space planning. By setting a maximum area of 175 square feet and establishing a relationship between length and width, the problem provides a framework for selecting a table that meets both functional and spatial requirements. Solving the inequality x(x + 18) ≤ 175 will yield the range of possible widths, but the final decision should also take into account the specific needs and preferences of the office environment. This careful consideration ensures that the conference table serves as a valuable asset for communication and collaboration within the office.
H2 Setting Up the Equation for Possible Widths
Establishing the equation is a crucial step in determining the possible widths of Mr. Nathan's conference table. We know the area constraint and the relationship between the length and width. To reiterate, the area of the conference table must not exceed 175 square feet, and the length is 18 feet more than the width (x). Therefore, we can formulate an equation that encapsulates these conditions, allowing us to solve for the possible values of x. This process involves translating the word problem into a mathematical expression that can be manipulated to find the desired solution.
Translating Words into Math
The first step in setting up the equation is to translate the given information into mathematical terms. We are told that the length of the table is 18 feet more than the width, x. This can be expressed as length = x + 18. The area of a rectangle is calculated by multiplying its length and width, so the area of the conference table can be represented as width × length = x(x + 18). Since the area must be no more than 175 square feet, we can write the inequality x(x + 18) ≤ 175. This inequality forms the foundation for solving the problem.
Expanding the Inequality
To solve the inequality, we first need to expand it. Multiplying x by (x + 18) gives us x² + 18x. So, the inequality becomes x² + 18x ≤ 175. This quadratic inequality is a standard form that can be manipulated to find the range of values for x that satisfy the condition. Understanding how to expand and simplify algebraic expressions is essential for solving this type of problem.
Rearranging the Inequality
Next, we need to rearrange the inequality to bring it into a standard quadratic form, which is ax² + bx + c ≤ 0. To do this, we subtract 175 from both sides of the inequality, resulting in x² + 18x - 175 ≤ 0. This form allows us to use various methods, such as factoring or the quadratic formula, to find the roots of the equation and determine the intervals where the inequality holds true. Rearranging the inequality is a key step in preparing it for further analysis.
Factoring the Quadratic Expression
One way to solve the quadratic inequality is by factoring the quadratic expression x² + 18x - 175. Factoring involves finding two numbers that multiply to -175 and add to 18. These numbers are 25 and -7. Therefore, the quadratic expression can be factored as (x + 25)(x - 7). So, the inequality becomes (x + 25)(x - 7) ≤ 0. Factoring simplifies the inequality and makes it easier to identify the critical points that define the intervals of possible solutions.
Identifying Critical Points
The critical points are the values of x that make the expression (x + 25)(x - 7) equal to zero. These points are found by setting each factor equal to zero and solving for x. This gives us x + 25 = 0 and x - 7 = 0, which yield x = -25 and x = 7. These critical points divide the number line into three intervals: x < -25, -25 < x < 7, and x > 7. The sign of the quadratic expression within each interval determines whether the inequality is satisfied. Identifying these critical points is essential for determining the feasible range of widths for the conference table.
Importance of the Non-Negative Width
In the context of the problem, the width of the conference table cannot be negative. Therefore, we can disregard the critical point x = -25 and any values of x less than -25. This practical consideration narrows the range of possible solutions and focuses our attention on the interval where the width is physically meaningful. Understanding the real-world constraints of the problem is crucial for interpreting the mathematical results.
Conclusion
Setting up the equation x² + 18x - 175 ≤ 0 is a critical step in solving the problem of determining the possible widths of Mr. Nathan's conference table. By translating the given information into mathematical terms, expanding and rearranging the inequality, factoring the quadratic expression, and identifying the critical points, we can establish the foundation for finding the feasible range of widths. Considering the practical constraint that the width must be non-negative, we narrow our focus to the relevant interval. This rigorous process ensures that the solution accurately reflects the problem's conditions and provides a meaningful answer.
H3 Solving the Inequality and Finding the Interval
Solving the inequality x² + 18x - 175 ≤ 0 is a critical step in determining the feasible range for the width (x) of Mr. Nathan's conference table. We've already factored the quadratic expression and identified the critical points. Now, we need to analyze the intervals defined by these critical points to find the interval(s) where the inequality holds true. This involves testing values within each interval to determine the sign of the quadratic expression, which will ultimately lead us to the solution.
Analyzing the Intervals
The critical points x = -25 and x = 7 divide the number line into three intervals: (-∞, -25), (-25, 7), and (7, ∞). To determine where the inequality (x + 25)(x - 7) ≤ 0 is satisfied, we need to test a value from each interval in the factored inequality.
Testing the Interval (-∞, -25)**
Let's choose a test value from this interval, such as x = -26. Plugging this value into the factored inequality, we get (-26 + 25)(-26 - 7) = (-1)(-33) = 33. Since 33 is not less than or equal to zero, the inequality is not satisfied in this interval.
Testing the Interval (-25, 7)**
Next, let's test a value from this interval, such as x = 0. Plugging this value into the factored inequality, we get (0 + 25)(0 - 7) = (25)(-7) = -175. Since -175 is less than or equal to zero, the inequality is satisfied in this interval. This indicates that widths within this interval meet the area constraint.
Testing the Interval (7, ∞)**
Finally, let's test a value from this interval, such as x = 8. Plugging this value into the factored inequality, we get (8 + 25)(8 - 7) = (33)(1) = 33. Since 33 is not less than or equal to zero, the inequality is not satisfied in this interval.
Determining the Solution Interval
From our analysis, we find that the inequality (x + 25)(x - 7) ≤ 0 is only satisfied in the interval (-25, 7). However, we must also consider the endpoints of the interval. Since the inequality includes “less than or equal to,” the endpoints x = -25 and x = 7 are also included in the solution.
Considering the Non-Negative Width
As we discussed earlier, the width of the conference table cannot be negative. Therefore, we need to consider only the non-negative portion of the solution interval. This means we disregard any values of x less than zero. The intersection of the interval (-25, 7) and the non-negative constraint gives us the interval [0, 7]. This interval represents the possible widths of the conference table that satisfy the area constraint.
Expressing the Solution
The possible widths of the conference table can be expressed as 0 ≤ x ≤ 7. This means that the width x must be greater than or equal to zero and less than or equal to 7 feet. This range ensures that the table's area does not exceed 175 square feet while adhering to the physical constraints of the problem.
Conclusion
Solving the inequality x² + 18x - 175 ≤ 0 and considering the non-negative width constraint has allowed us to determine the interval of possible widths for Mr. Nathan's conference table. By analyzing the intervals defined by the critical points and testing values within each interval, we found that the inequality is satisfied when 0 ≤ x ≤ 7. This range provides a practical guideline for selecting a conference table that meets the specified area requirement and fits comfortably within the office space. Understanding how to solve quadratic inequalities and apply real-world constraints is essential for problem-solving in various contexts.
H4 Possible Widths for the Conference Table
Determining the possible widths for the conference table in Mr. Nathan's office involves understanding the solution interval we derived from the inequality. The inequality x(x + 18) ≤ 175 represents the constraint that the area of the table must be no more than 175 square feet, where x is the width and x + 18 is the length. After solving this inequality and considering the non-negative width constraint, we found that the possible widths lie within the interval 0 ≤ x ≤ 7. This interval provides a range of values for the width that satisfy the given conditions.
Interpreting the Interval
The interval 0 ≤ x ≤ 7 means that the width (x) of the conference table can be any value between 0 and 7 feet, inclusive. A width of 0 feet would result in a degenerate table with no area, which is not practical. Therefore, while mathematically 0 is included in the interval, a more realistic interpretation would start slightly above 0 to allow for a functional table. The upper limit of 7 feet provides a clear boundary for the maximum width that the table can have while still meeting the area constraint.
Calculating the Corresponding Lengths
For each possible width within the interval 0 ≤ x ≤ 7, we can calculate the corresponding length using the relationship length = x + 18. For example, if the width is 0 feet, the length would be 18 feet. If the width is 7 feet, the length would be 25 feet. These pairs of dimensions define the possible sizes of the conference table that adhere to the specified area limit. Understanding how the width and length are related allows for a more comprehensive assessment of the table's dimensions.
Practical Implications of the Width Interval
The interval of possible widths has significant practical implications for selecting the right conference table. A narrower table might be suitable for smaller meetings or rooms where space is limited. A wider table, closer to the 7-foot limit, would provide more space for attendees and materials but might require a larger room to accommodate it comfortably. The choice of width should also consider the overall layout of the office and the desired aesthetic.
Choosing the Optimal Width
Selecting the optimal width for the conference table involves balancing the need for space and functionality with the constraints of the room and the budget. A width in the middle of the interval, such as 4 or 5 feet, might provide a good compromise between space efficiency and usability. Ultimately, the best width will depend on the specific needs and preferences of Mr. Nathan's office.
Alternative Representations of the Interval
The interval 0 ≤ x ≤ 7 can also be expressed in interval notation as [0, 7]. This notation uses square brackets to indicate that the endpoints 0 and 7 are included in the interval. Understanding different ways to represent the solution can help in communicating the results more effectively.
Conclusion
The possible widths for the conference table in Mr. Nathan's office are defined by the interval 0 ≤ x ≤ 7. This range ensures that the table's area does not exceed 175 square feet while considering the relationship between the width and length. By interpreting this interval and understanding its practical implications, we can make informed decisions about selecting a conference table that meets the needs of the office. This comprehensive analysis highlights the importance of mathematical problem-solving in real-world scenarios and the value of considering practical constraints when interpreting mathematical results.
H5 Selecting the Right Conference Table
Selecting the right conference table for Mr. Nathan's office involves considering a variety of factors, including the possible widths we've determined, the overall dimensions of the room, the number of regular attendees at meetings, and the intended use of the table. The goal is to choose a table that not only fits within the space and budget but also enhances the functionality and aesthetics of the office. This decision requires a careful evaluation of the needs and preferences of the office environment.
Considering the Room Dimensions
The dimensions of the room are a primary factor in determining the appropriate size and shape of the conference table. A table that is too large can make the room feel cramped and hinder movement, while a table that is too small may not provide enough space for attendees and materials. It's essential to measure the room accurately and consider the placement of other furniture and fixtures to ensure that the conference table fits comfortably within the space. This careful planning ensures that the table serves its purpose without compromising the room's usability.
Assessing the Number of Attendees
The number of people who typically attend meetings and discussions will influence the ideal size of the conference table. A larger table is necessary for accommodating more attendees, while a smaller table may be sufficient for smaller groups. It's important to consider both the typical number of attendees and any occasional larger meetings that may occur. Providing enough space for each attendee to comfortably work and interact is crucial for effective communication and collaboration.
Evaluating the Intended Use
The intended use of the conference table will also affect the optimal size and shape. If the table will primarily be used for formal meetings and presentations, a larger table with a formal design may be appropriate. If the table will be used for more informal discussions and collaborative work, a smaller, more flexible table may be preferable. Considering the specific activities that will take place at the table helps ensure that it meets the functional requirements of the office.
Considering the Shape of the Table
While we have focused on the width and length of a rectangular table, the shape of the table is another important consideration. Rectangular tables are a common choice for conference rooms, but other shapes, such as oval or round tables, may be more suitable for certain spaces or purposes. Oval tables can promote better communication among attendees, while round tables can foster a more collaborative atmosphere. The choice of shape should align with the office's culture and communication style.
Budget Considerations
The budget for the conference table is a practical constraint that must be taken into account. Conference tables can range in price from a few hundred dollars to several thousand dollars, depending on the size, materials, and design. Setting a budget early in the selection process helps narrow the options and ensures that the final choice is financially feasible. Balancing cost with quality and functionality is essential for making a wise investment.
Aesthetics and Design
The aesthetics and design of the conference table should complement the overall style of the office. The table should be visually appealing and contribute to a professional and welcoming environment. Consider the materials, finish, and design elements of the table and how they align with the office's décor. A well-designed conference table can enhance the overall impression of the office and create a positive atmosphere for meetings and discussions.
Conclusion
Selecting the right conference table for Mr. Nathan's office is a multifaceted decision that involves considering the possible widths, room dimensions, number of attendees, intended use, shape, budget, and aesthetics. By carefully evaluating these factors and aligning the choice with the needs and preferences of the office, a conference table can be selected that not only meets the functional requirements but also enhances the overall environment. This comprehensive approach ensures that the investment in a conference table is a valuable one that supports effective communication and collaboration within the office.
