Question

Certain instruments used in veterinary surgery can be made either of stainless steel or of nylon. In a study of such instruments, 50 complete sterilizations of a set of nylon instruments required \(3.4\) times the amount of energy used to manufacture that set of instruments, whereas 50 complete sterilizations of a set of stainless steel instruments required \(2.1\) times the amount of energy required to manufacture that set of instruments. If the statements above are true, each of the following could be true EXCEPT: (A) The 50 complete sterilizations of the nylon instruments used more energy than did the 50 complete sterilizations of the stainless steel instruments. (B) More energy was required for each complete sterilization of the nylon instruments than was required to manufacture the nylon instruments. (C) More nylon instruments than stainless steel instruments were sterilized in the study. (D) More energy was used to produce the stainless steel instruments than was used to produce the nylon instruments. (E) The total cost of 50 complete sterilizations of the stainless steel instruments was greater than the cost of manufacturing the stainless steel instruments.

Short answer

Option (B) cannot be true.

Step-by-step solution

Understanding the Proportions

We are given that sterilizing nylon instruments takes 3.4 times the manufacturing energy, while sterilizing stainless steel instruments takes 2.1 times the manufacturing energy. Essentially, if the energy required to manufacture nylon is \( E_n \) and for stainless steel is \( E_s \), then the energy for 50 sterilizations of nylon is \( 3.4E_n \), and for stainless steel is \( 2.1E_s \).

Analyze Option (A)

Option (A) suggests that sterilizing nylon uses more energy than sterilizing stainless steel, i.e., \( 3.4E_n > 2.1E_s \). There is a possibility this is true if the manufacturing energy of nylon \( E_n \) is reasonably higher than \( E_s \) or if their ratio fits certain values.

Analyze Option (B)

Option (B) states that each complete sterilization of nylon uses more energy than manufacturing nylon itself. If true, \( \frac{3.4E_n}{50} > E_n \) simplifies to \( 3.4 > 50 \), which is false. Thus, sterilization does not use more energy per instance than manufacturing does.

Analyze Option (C)

Option (C) states more nylon instruments than stainless steel instruments were sterilized. Nothing in the problem provides a direct measure to count instruments, so this is possible depending on relative choices in amounts sterilized.

Analyze Option (D)

Option (D) suggests producing stainless steel uses more energy than nylon, i.e., \( E_s > E_n \). This may or may not be true since we lack direct energy cost values but is plausible depending on efficiencies.

Analyze Option (E)

Option (E) assesses that the cost of sterilizing stainless is higher than production cost \( 2.1E_s > E_s \). Therefore, \( 2.1 > 1 \) is trivially true, showing sterilization uses more total energy than manufacturing independently these instruments.

Identify the False Statement

After reviewing each statement, option (B) is the only one that arrives at a mathematical contradiction, as the energy per sterilization cannot exceed the manufacturing one by their ratio.

Key concepts

Energy Consumption

Energy consumption plays a crucial role in decision-making, both for economic and environmental reasons. In the context of this exercise, evaluating the energy used in the sterilization and manufacturing process for veterinary surgery instruments is important.

When comparing energies, you'll often see terms like "times the energy," meaning how much more energy one process uses compared to another. For instance, if sterilization uses "3.4 times" the energy it takes to manufacture the instruments, this signals that the sterilization process is much more energy-intensive than production. It's essential to grasp this multiplier effect as it directly impacts costs and resource management.

By understanding energy consumption in devices, such as the veterinary tools in this exercise, we gain insights into optimizing both manufacturing and operating costs. This encourages advancements in techniques and materials, potentially lowering energy needs in the future.

Veterinary Surgery

Veterinary surgery involves various procedures necessary for the health and well-being of animals. Instruments used in these surgeries are critical, and their preparation, including sterilization, must comply with stringent sanitary standards.

Instruments made from different materials can behave differently. For instance, stainless steel is durable and generally more resistant to damage during sterilization. Nylon, while lighter and sometimes cheaper, may not endure as many sterilization cycles, leading to higher energy costs as reflected in the exercise data.

Different surgeries require specific instruments, and veterinary surgeons choose materials based on the intended purpose and the animals being treated. A deeper understanding of these instruments' properties can aid in better choice, ensuring effective surgeries while considering the environmental and energy costs.

Instrument Sterilization

Sterilization of instruments is essential in any surgical practice to prevent infections. This process requires the application of heat, chemicals, or radiation, all of which demand significant energy.

In the exercise, it's shown that nylon instruments require more energy for sterilization compared to their stainless steel counterparts. This could be due to the lower thermal tolerance of nylon, necessitating more careful, and energy-intensive cycles to ensure complete sterilization without damage.

Efficiency in sterilization processes is critical to minimize not only energy consumption but also to prolong the lifespan of the instruments. Through careful selection of materials and process optimization, veterinary practices can effectively manage their environmental impact and reduce operational costs over time.

Mathematical Contradiction

A mathematical contradiction occurs when an assumption or statement negates itself or leads to an impossible conclusion. In logical reasoning exercises, identifying contradictions is key to solving problems.

In the given exercise, option (B) presents a contradiction by claiming each sterilization requires more energy than the manufacturing of the instrument itself. Simplifying gives an incorrect inequality \(3.4/50 > 1\), suggesting sterilization per instance uses more energy than producing one instrument, which is false as sterilization percentage exceeds that of production.

Identifying such contradictions helps in refining arguments and developing better problem-solving skills. Recognizing that a statement leads to an impossible outcome is a powerful tool to understand the underlying logical framework of any problem, assisting students and professionals alike in making reasoned decisions.