Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 37-38: Ozone occurs at all levels of Earth's atmosphere. The density of ozone varies both seasonally and latitudinally. At Edmonton, Canada, the density \(D(h)\) of ozone (in \(10^{-3} \mathrm{~cm} / \mathrm{km}\) ) for altitudes \(h\) between 20 kilometers and 35 kilometers was determined experimentally. For each \(D(h)\) and season, approximate the altitude at which the density of ozone is greatest. \(D(h)=-0.058 h^{2}+2.867 h-24.239\) (autumn)

Step-by-Step Solution

Answer

The ozone density is greatest at approximately 24.71 km altitude.

Step 1: Understand the Goal

Our task is to find the altitude, \( h \), at which the density of ozone, \( D(h) \), is greatest between altitudes 20 km and 35 km. This means we need to find the maximum point of the given quadratic function.

Step 2: Identify the Function Type

The given function \( D(h) = -0.058h^2 + 2.867h - 24.239 \) is a quadratic function in terms of \( h \). Since the coefficient of \( h^2 \) is negative (\( -0.058 \)), the parabola opens downwards, indicating a maximum point at its vertex.

Step 3: Find the Vertex of the Quadratic Function

The vertex form of a parabola, which is \( ax^2 + bx + c \), reaches its maximum or minimum at the vertex, given by \( h = -\frac{b}{2a} \). For our function, substitute \( a = -0.058 \) and \( b = 2.867 \) to find the altitude:\[h = -\frac{2.867}{2 \times (-0.058)} = \frac{2.867}{0.116} \approx 24.71 \text{ km}\]

Step 4: Verify the Altitude

The calculated vertex, \( h = 24.71 \) km, lies within the given range of 20 km to 35 km. Thus, it is the valid altitude where the ozone density is greatest.

Key Concepts

Understanding Maximum Point in Quadratic Functions

When dealing with quadratic functions, finding the point where the function reaches its highest value, or maximum point, is crucial. In the quadratic function \[D(h) = -0.058h^2 + 2.867h - 24.239\]we see a typical example. Here, the coefficient of \(h^2\) is negative. This is key information. A negative coefficient means the parabola opens downward, indicating that there is a maximum, rather than a minimum, point somewhere on the curve.
To find this peak, we use the concept of a vertex, which provides the maximum or minimum point of the quadratic function. For functions shaped like an upside-down "U", finding the vertex is equal to finding the maximum point.
To compute it: identify the appropriate values from the equation (here, \(a = -0.058\) and \(b = 2.867\)). Then, calculate the vertex \(h = -\frac{b}{2a}\). Understanding this mathematical behavior helps us predict where certain values, such as maximum density, occur in various scenarios like atmospheric studies.

The Role of Vertex Form in Quadratics

Vertex form is an alternate way to represent the typical quadratic function \(ax^2 + bx + c\). It yields significant insights about the function's graph. Unlike the standard form, vertex form easily reveals the vertex, a critical point in applications such as maximization problems.
The vertex form of a quadratic equation is \(a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola. This form allows us to see instantly the peak or the lowest point of the curve because the vertex is clearly shown. It is highly useful for finding max or min points without completing the square directly.
In our specific function \(D(h) = -0.058h^2 + 2.867h - 24.239\), the vertex is found by calculating \(h = -\frac{b}{2a}\). This doesn't transform our equation into vertex form but helps us identify important values similar to what the vertex form would show, such as the maximum point and location.

Altitude Calculation and its Importance

Altitude calculation is a vital tool in understanding atmospheric behaviors, such as ozone concentration. The formula \(h = -\frac{b}{2a}\) is used to determine the specific altitude \(h\) where a function, such as ozone density \(D(h)\), reaches its highest value. This calculation is performed by using the vertex formula of a quadratic function.
In situations where researchers study atmospheric layers, knowing the altitude that maximizes certain conditions, like ozone concentration, is essential. It can help with climate models, weather predictions, and understanding environmental changes.
In the given scenario, after computing \(h = \frac{2.867}{0.116} \approx 24.71\, km\), we discovered the peak ozone density occurs around 24.71 km altitude. This insight is crucial for scientists looking to map environmental patterns, offering precise data to maintain ecological balance and promoting safer, healthier environments.