Problem 25 - Discrete Mathematics with Applications

Question

There are seven lots, 1 through \(7,\) to be developed in a certain city. A builder would like to build one bank, two hotels, and two restaurants on these lots, subject to the following restrictions by the city planning board (The Official LSAT PrepBook, 1991): If lot 2 is used, lot 4 cannot be used. If lot 5 is used, lot 6 cannot be used. The bank can be built only on lot \(5,6,\) or \(7 .\) A hotel cannot be built on lot \(5 .\) A restaurant can be built only on lot \(1,2,3,\) or \(5 .\) Which of the following is a possible list of locations for building them? A. The bank on lot \(7,\) hotels on lots 1 and \(4,\) and restaurants on lots 2 and 5 B. The bank on lot \(7,\) hotels on lots 3 and \(4,\) and restaurants on lots 1 and 5 C. The bank on lot \(7,\) hotels on lots 4 and \(5,\) and restaurants on lots 1 and 3.

Step-by-Step Solution

Answer

The possible list of locations for building them is the bank on lot 7, hotels on lots 3 and 4, and restaurants on lots 1 and 5.

Step 1: Option A

The bank on lot 7, hotels on lots 1 and 4, and restaurants on lots 2 and 5. - Bank on lot 7 satisfies condition 3. - Hotels on lots 1 and 4 do not violate condition 4. - Restaurants on lots 2 and 5 satisfy condition 5. - Since lot 2 is used, lot 4 cannot be used according to condition 1. But lot 4 is used for a hotel. So, this option is not valid.

Step 2: Option B

The bank on lot 7, hotels on lots 3 and 4, and restaurants on lots 1 and 5. - Bank on lot 7 satisfies condition 3. - Hotels on lots 3 and 4 do not violate condition 4. - Restaurants on lots 1 and 5 satisfy condition 5. - No restrictions on using lots 2, 4, 5, or 6 are violated. So, this option is valid.

Step 3: Option C

The bank on lot 7, hotels on lots 4 and 5, and restaurants on lots 1 and 3. - Bank on lot 7 satisfies condition 3. - Hotels on lots 4 and 5 violate condition 4, as a hotel cannot be built on lot 5. - Restaurants on lots 1 and 3 satisfy condition 5. - Since a hotel is built on lot 5 which violates condition 4, this option is not valid. Since option B is the only valid choice, the answer is: The bank on lot 7, hotels on lots 3 and 4, and restaurants on lots 1 and 5.

Key Concepts

Lot allocation

In urban planning, especially in city development projects, lot allocation is a significant process. It involves deciding which structures will be built on specific pieces of land. When working with multiple lots, like in the problem with seven lots, it's important to adhere to certain rules.
Lot allocation considers factors like the type of building and its appropriate placement. For instance, a bank might need a secure location, while restaurants might prefer high foot traffic areas.
In our exercise, the placement had to follow the city's restrictions:

• Banks can only be on lots 5, 6, or 7.
• Hotels cannot occupy lot 5.
• Restaurants have options between lots 1, 2, 3, and 5.

This means that each type of building has designated zones, ensuring they meet city planning codes and are fit for purpose, enhancing city life and functionality.

Building restrictions

To ensure safety, functionality, and community welfare, city planning boards impose building restrictions. These are rules that developers must follow when deciding how to use each lot.
In the given scenario, restrictions are based on potential conflicts between lots. For example, if lot 2 is used, lot 4 cannot be, due to overlap in infrastructure or safety concerns. Similarly, using lot 5 precludes using lot 6.

• Such constraints regulate density, preventing excessive clustering of certain types of buildings.
• They also preserve scenery and access rights, balancing development.
  

Developers need to factor in these restrictions during the planning phase to ensure compliance and minimize legal hurdles. Proper adherence can also impact the community positively by maintaining an organized urban environment.

Potential configurations

Potential configurations refer to the different layouts or arrangements that satisfy the given building restrictions and objectives. It's like solving a puzzle where each piece must fit perfectly without overriding another's space or condition.
In the provided problem, permissible configurations included different combinations of lots for banks, hotels, and restaurants. Option B proved valid:

• Placing the bank on lot 7 ensured adherence to available choices for banks.
• Hotels on lots 3 and 4 were permissible since they did not contradict building lot rules.
• Restaurants on lots 1 and 5 fit neatly within the designated lots.

Configurations often rely on trial and error within defined parameters, requiring close attention to detail. This process ensures the efficient use of space, adherence to regulations, and optimized layout for practical and aesthetic purposes.