Problem 53 - Algebra and Trigonometry with Analytic Geometry

Question

Ventilation is an effective way to improve indoor air quality. In nonsmoking restaurants, air circulation requirements (in \(\mathrm{ft}^{3} / \mathrm{min}\) ) are given by the function \(V(x)=35 x\), where \(x\) is the number of people in the dining area. (a) Determine the ventilation requirements for 23 people. (b) Find \(V^{-1}(x)\). Explain the significance of \(V^{-1}\). (c) Use \(V^{-1}\) to determine the maximum number of people that should be in a restaurant having a ventilation capability of \(2350 \mathrm{ft}^{3} / \mathrm{min}\).

Step-by-Step Solution

Answer

(a) 805 \(\mathrm{ft}^3/\mathrm{min}\); (b) \(V^{-1}(x) = \frac{x}{35}\), finds people from ventilation; (c) 67 people.

Step 1: Calculate Ventilation for 23 People

To find the ventilation requirements for 23 people, substitute \(x = 23\) into the function \(V(x) = 35x\). This gives:\[V(23) = 35 \times 23 = 805 \, \mathrm{ft}^3/\mathrm{min}\]Thus, the ventilation requirement for 23 people is 805 \(\mathrm{ft}^3/\mathrm{min}\).

Step 2: Find the Inverse Function \(V^{-1}(x)\)

To find the inverse of \(V(x) = 35x\), solve for \(x\) in terms of \(V\):\[x = \frac{V}{35}\]Thus, the inverse function is \(V^{-1}(x) = \frac{x}{35}\). The significance of \(V^{-1}(x)\) is that it allows you to determine the number of people in the dining area given the ventilation capacity.

Step 3: Use \(V^{-1}(x)\) to Determine Maximum People for Given Ventilation

To find the maximum number of people for a ventilation capacity of 2350 \(\mathrm{ft}^3/\mathrm{min}\), use \(V^{-1}(x) = \frac{x}{35}\) and substitute \(x = 2350\):\[V^{-1}(2350) = \frac{2350}{35} = 67\]Thus, the maximum number of people that can be in a restaurant with a ventilation capability of 2350 \(\mathrm{ft}^3/\mathrm{min}\) is 67.

Key Concepts

Ventilation Requirements

Ventilation requirements are crucial when it comes to ensuring a healthy indoor environment, particularly in places like restaurants. Ventilation refers to the supply of fresh air to a space, eliminating the stale air, and thereby improving the air quality.

• In non-smoking restaurants, the amount of air needed is often determined by the number of occupants.
• Understanding these requirements helps ensure that everyone inside is comfortable and breathing quality air.

In this context, mathematical formulas are handy. For example, the function given in the exercise, \(V(x) = 35x\), helps compute the necessary cubic feet per minute of air needed, based on the number of people.
This function makes it simple to calculate exactly how much air circulation is required, ensuring environments meet health standards.

Air Circulation

Air circulation is vital for maintaining indoor air quality, as it helps disperse airborne pollutants and provides fresh air, which is especially significant in shared spaces like restaurants.

• Proper air circulation reduces the buildup of carbon dioxide and volatile organic compounds.
• It helps regulate temperature and humidity, creating a more comfortable environment for occupants.

In the problem given, air circulation requirements are calculated as a product of the number of people, exemplified by the equation \(V(x) = 35x\). This practical approach ensures that enough fresh air enters the dining area based on its occupancy.

Mathematical Modeling

Mathematical modeling is a fantastic method to simulate real-world processes, like determining ventilation requirements, with precision and efficiency.

• It involves forming equations or functions that describe relationships between different variables.
• The model enables predictions and solutions for varying scenarios.

In the given scenario, the mathematical model \(V(x) = 35x\) represents how air requirements rise with an increase in the number of people.
Using this model, calculations can be easily adjusted. Just plug in the number of occupants to predict the needed ventilation, making it a dynamic tool for real-time application.

Algebraic Manipulation

Algebraic manipulation helps transform and solve mathematical equations by rearranging terms and solving for unknowns.

• It is used here to find the inverse of the function \(V(x) = 35x\), which is \(V^{-1}(x) = \frac{x}{35}\).
• This inverse function is crucial for determining how many people can be accommodated based on specific ventilation limits.

For instance, with a given ventilation capability, say 2350 \( \mathrm{ft}^3/\mathrm{min} \), we can use algebraic manipulation of the inverse equation \(V^{-1}(x)\) to calculate that the space can hold up to 67 people.
Thus, algebraic manipulation extends the utility of our mathematical model to explore various practical solutions.