Question

the amount of a particular pollutant ( p ) in the air depends on the wind speed ( s ), among other things, with the relationship between ( p ) and ( s ) approximated by ( p = 25-0.02s^{2} ), where ( p ) is in ounces per cubic yard and ( s ) is in miles per hour. a. sketch the graph of this model with ( s ) on the horizontal axis and with nonnegative values of ( s ) and ( p ). b. is the function increasing or decreasing on this domain? c. what is the ( p )-intercept of the graph? d. what does the ( p )-intercept mean in the context of this application?

Explanation:

Step1: Analyze the function form

The function \(p = 25-0.02s^{2}\) is a quadratic function of the form \(y = ax^{2}+bx + c\) (here \(a=- 0.02\), \(b = 0\), \(c = 25\)). Since \(a=-0.02<0\), the parabola opens downwards. Also, when \(s = 0\), \(p=25\).

Step2: Check the domain for the application

The wind - speed \(s\geq0\) (non - negative values as per the problem statement). As \(s\) increases from \(0\) to \(\infty\), for the function \(p(s)=25 - 0.02s^{2}\), we can use the derivative (or analyze the behavior of the quadratic function). The derivative \(p^\prime(s)=-0.04s\). For \(s>0\), \(p^\prime(s)<0\).

Step3: Find the \(p\) - intercept

To find the \(p\) - intercept, we set \(s = 0\) in the function \(p(s)\). Substituting \(s = 0\) into \(p=25-0.02s^{2}\), we get \(p=25\).

Answer:

a. The correct graph is D (since when \(s = 0\), \(p = 25\) and the function is a downward - opening parabola for \(s\geq0\)).
b. The function \(p(s)\) is decreasing on the domain \(s\geq0\) (because \(p^\prime(s)=-0.04s<0\) for \(s > 0\)).
c. The \(p\) - intercept is \(25\).
d. In the context of this application, when the wind speed \(s = 0\) miles per hour, the amount of the pollutant \(p\) in the air is \(25\) ounces per cubic yard.