Altitude and Pressure At an altitude of h feet the...

Altitude and Pressure At an altitude of $h$ feet the atmospheric pressure $a$ (atm) is given by $$ a=3.89 \times 10^{-10} h^{2}-3.48 \times 10^{-5} h+1 $$ (Sportscience, www.sportsci.org). a. Graph this quadratic function. b. $\ln 1953,$ Edmund Hillary and Tenzing Norgay were the first persons to climb Mount Everest (see accompanying figure). Was the atmospheric pressure increasing or decreasing as they went to the $29,029$ -ff summit? c. Where is the function decreasing? Increasing? d. Does your answer to part (c) make sense? e. For what altitudes do you think the function is valid?

Altitude and Pressure At an altitude of $h$ feet the atmospheric pressure $a$ (atm) is given by
$$
a=3.89 \times 10^{-10} h^{2}-3.48 \times 10^{-5} h+1
$$
(Sportscience, www.sportsci.org).
a. Graph this quadratic function.
b. $\ln 1953,$ Edmund Hillary and Tenzing Norgay were the first persons to climb Mount Everest (see accompanying figure). Was the atmospheric pressure increasing or decreasing as they went to the $29,029$ -ff summit?
c. Where is the function decreasing? Increasing?
d. Does your answer to part (c) make sense?
e. For what altitudes do you think the function is valid?

Key Concepts

Application of Mathematical Models in Physical Sciences

Mathematical models, such as the quadratic representation of atmospheric pressure with altitude, help in approximating real-world phenomena. These models allow us to predict and analyze the behavior of physical systems, but they often come with limitations and assumptions that must be acknowledged. Understanding how to interpret the parameters and behavior of the model is key to applying it effectively in a scientific context.

Increasing and Decreasing Intervals

Determining the intervals where a function is increasing or decreasing is essential for understanding its behavior over the domain. For quadratic functions, the vertex divides the domain into two intervals: one where the function is monotonically increasing and one where it is decreasing. This concept is useful not only in pure mathematics but also in practical applications where the rate of change is significant.

Domain and Validity of Models

Every mathematical model has a specific domain within which it accurately describes the observed phenomena. Recognizing the appropriate range of input values—such as altitude ranges where atmospheric pressure can be effectively modeled by a quadratic function—is important. Beyond this domain, the model may lose its validity and no longer reflect the true physical behavior.

Quadratic Functions

A quadratic function is a polynomial of degree two, typically written in the form ax² + bx + c, where a, b, and c are constants. Its graph is a parabola, which either opens upward if a is positive or downward if a is negative. Understanding the standard form of a quadratic function is crucial for analyzing various features such as the vertex, axis of symmetry, and the direction of the opening.

Graphing Quadratic Functions

Graphing a quadratic function involves plotting its parabola by identifying key features such as the vertex, intercepts, and the axis of symmetry. The vertex is found using the formula -b/(2a), and it serves as a turning point for the function. Graphical analysis helps in visualizing the behavior of the function over its domain and is a vital step in interpreting the physical meaning of the function’s parameters.

Transcript

00:01 So the quadratic that they give us tells us the atmospheric pressure based off the altitude in feet.

00:10 And we're first asked to graph this quadratic equation.

00:13 So i went ahead and graft it already, but if you want to actually do this by hand as opposed to just plug it into a calculator, i'll talk about how we can do that as well.

00:21 So the only point we really need to find is the vertex.

00:25 Because we know over here, since this is 1, or our y intercept, we should start from there, and then we would just kind of go from there.

00:37 So to find the vertex, what we're going to do is use the formula, h is equal to negative b over 2, and i'm going to use parentheses for our quotations for this, because this is a different a than our atmospheric pressure.

00:57 Because remember we would write this quadratic as this term here is going to be a and then this term here is going to be b well let's go ahead and plug all this in to figure out what altitude this occurs at this minimum pressure so it would be negative 3 .48 times 10 to the negative times this negative 5th all over 3 .89 times 10 to the negative 10th and simplifying that all down should give us 17 ,400 ,000 over 3 which is approximately equal to 4 ,473 which is exactly what we have there and then to get the output you would just plug this value in for h to get that it should be something around 0 .22.

02:03 So that's how we would get the graph.

02:05 And then to get our other point, we would know that since our function is symmetric about the interval, we would just need to go double this distance out to reach another value where this is going to be equal to 1.

02:23 So that's just how we could graph this here.

02:29 So now that we have the graph, they want us to determine if the pressure is increasing or decreasing, at the summit of about 29 ,000.

02:39 So, well, 29 ,000 is going to be around here somewhere.

02:43 And if we were to look at this, it's definitely decreasing.

02:47 So it should be decreasing.

02:52 Now, it says, where is the function increasing? where is it decreasing? well, it should be increasing to the left of our vertex, or decreasing to the left of our vertex, looking at our graph over here.

03:06 And so it doesn't make sense to really, talk about, well, i guess it does, makes sense to talk about going below sea level, but in most cases, we're not going to...

Please give Ace some feedback