The rate of ventilation required in a public school...

The rate of ventilation required in a public school classroom depends on the volume of air space per child. The model $$y=80.4-11 \ln x, \quad 100 \leq x \leq 1500$$ approximates the minimum required rate of ventilation $y$ (in cubic feet per minute per child) in a classroom with $x$ cubic feet of air space per child. a. Use a graphing utility to graph the function and approximate the required rate of ventilation in a room with 300 cubic feet of air space per child. b. A classroom of 30 students has an air conditioning system that moves 450 cubic feet of air per minute. Determine the rate of ventilation per child. c. Use the graph in part (a) to estimate the minimum required air space per child for the classroom in part (b). d. The classroom in part (b) has 960 square feet of floor space and a ceiling that is 12 feet high. Is the rate of ventilation for this classroom adequate? Explain.

The rate of ventilation required in a public school classroom depends on the volume of air space per child. The model
$$y=80.4-11 \ln x, \quad 100 \leq x \leq 1500$$
approximates the minimum required rate of ventilation $y$ (in cubic feet per minute per child) in a classroom with $x$ cubic feet of air space per child.
a. Use a graphing utility to graph the function and approximate the required rate of ventilation
in a room with 300 cubic feet of air space per child.
b. A classroom of 30 students has an air conditioning system that moves 450 cubic feet of air per minute. Determine the rate of ventilation per child.
c. Use the graph in part (a) to estimate the minimum required air space per child for the classroom in part (b).
d. The classroom in part (b) has 960 square feet of floor space and a ceiling that is 12 feet high. Is the rate of ventilation for this classroom adequate? Explain.

Key Concepts

Volume Calculation

Volume calculation involves determining the amount of space within a three-dimensional object by multiplying its dimensions, such as floor area by height. This concept is vital when assessing requirements that depend on the amount of available space, like ventilation in a room.

Function Evaluation

Function evaluation is the process of substituting a given input value into a function to determine its corresponding output. This is a basic but crucial concept used to compute specific outcomes from a mathematical model by direct substitution.

Ratios and Proportions

Ratios and proportions are used to compare quantities by dividing one quantity by another. In applied scenarios, such as determining per-person rates, these calculations help distribute a total amount evenly among individuals, which is essential in many real-world applications.

Logarithmic Functions

Logarithmic functions involve the natural logarithm (ln) and are useful in modeling relationships where a rate of change decreases as the independent variable increases. They help describe processes that exhibit diminishing returns and are fundamental in various scientific and engineering contexts.

Graphical Analysis of Functions

Graphical analysis involves plotting functions to visually examine their behavior, evaluate specific values, and identify trends. This method is particularly useful for understanding how changes in input affect output, as well as for making estimations based on the shape and features of the graph.

Transcript

00:01 Okay, so we're taking a look at an application of natural law here.

00:04 And right now we're looking at the rate of ventilation required in a public school classroom depending on the volume of airspace per child, which is going to be important down the line here.

00:13 So the first thing that we want to be able to do in question a is we want to take this information and we want to use our graph and be able to figure out how much ventilation would be necessary for 300 cubic feet per children for child.

00:26 So in part a here, what we're going to do is we're going to do is we're going to let x be equal to 300 cubic feet per child okay and when we plug that into our calculator we can use our graph and calculate graphing our graph to approximate that value and what we're going to find out is we're going to need about 17 .66 cubic feet per minute of ventilation per child in order to have the proper ventilation so in the second part, what we want to take a look at is we have a classroom of 30 children, and there's 450 cubic feet per minute would be the expectation.

01:16 When we divide that, that tells me that we must have at least 15 cubic feet per minute per child in order to be acceptable.

01:26 Well, we're going to need to use this in order to find out whether we're going to have an adequate amount of space based on the last problem.

01:33 So what we have now is we have this classroom, and it's this rectangular prism of a classroom.

01:41 My draw is not spectacular, but the base of my classroom is 960 square feet, and the height of the classroom is 12 feet.

01:53 When i multiply those together, i can find the volume of the entire classroom...

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