Problem 38 - 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. \(D(h)=-0.078 h^{2}+3.811 h-32.433\) (spring)

Step-by-Step Solution

Answer

The maximum ozone density occurs at approximately 24.43 kilometers altitude.

Step 1: Understand the Problem

We need to find the altitude, \(h\), for the maximum density of ozone described by the quadratic function \(D(h) = -0.078h^2 + 3.811h - 32.433\). This function is applicable for \(h\) values between 20 kilometers and 35 kilometers.

Step 2: Identify the Maximum Point of a Quadratic Function

A quadratic function \(ax^2 + bx + c\) reaches its maximum or minimum at \(x = -\frac{b}{2a}\). In this case, since the coefficient of \(h^2\) is negative (-0.078), it is an upside-down parabola, which means it will reach a maximum value at this point.

Step 3: Calculate the Vertex

Substitute the coefficients \(a = -0.078\) and \(b = 3.811\) into the formula for the x-coordinate of the vertex:\[h = -\frac{3.811}{2 \times -0.078} = \frac{3.811}{0.156} \approx 24.43\!\]The vertex at \(h = 24.43\) kilometers is within the specified range (20 to 35 kilometers).

Step 4: Conclusion on Altitude for Maximum Density

Since \(h = 24.43\) is within the specified altitude range and gives us the maximum of the quadratic function, the density of ozone is greatest at approximately 24.43 kilometers in altitude during spring at Edmonton, Canada.

Key Concepts

Vertex Formula

The vertex formula is a key part of understanding quadratic functions like the one used to determine the density of ozone at various altitudes. A quadratic function is generally represented as \(ax^2 + bx + c\). The vertex formula helps find the highest or lowest point of these functions, known as the vertex.

For any quadratic function, the x-coordinate of the vertex, which represents the point of maximum (or minimum) value, can be found using the vertex formula: \[x = -\frac{b}{2a}\]

This is extremely useful because it tells us where the function reaches its peak (or dip). In scenarios like the one in Edmonton where the upside-down parabola represents ozone density, employing the vertex formula helps us determine at which altitude the ozone density is the greatest.

Maximum Value

In quadratic functions, determining the maximum value is particularly straightforward when the function's graph forms a downward-facing parabola. This is because the highest point of the graph, the vertex, represents this maximum value.

The given function for ozone density at Edmonton is \(D(h) = -0.078h^2 + 3.811h - 32.433\). Here, the negative coefficient of the \(h^2\) term tells us that the parabola opens downwards. To find this maximum density value, we first calculate the x-coordinate of the vertex using the formula \(-\frac{b}{2a}\), which here represents the altitude \(h\).

In our example, substituting \(a = -0.078\) and \(b = 3.811\), we calculate:

• \(h = -\frac{3.811}{2 \times -0.078} = \frac{3.811}{0.156} \approx 24.43\)

Here, the altitude of approximately 24.43 kilometers is where the density of ozone reaches its peak during spring in Edmonton.

Parabola

A parabola is the curve produced when graphing quadratic functions. It can open upwards or downwards depending on the coefficient of the \(x^2\) term in the quadratic equation. This shape is central to analyzing the behavior of such functions.

For the given ozone density function \(D(h) = -0.078h^2 + 3.811h - 32.433\), the "\(h^2\)" term has a negative coefficient (\(-0.078\)), which means the parabola is upside-down.

In an upside-down parabola:

• The function has a maximum point (vertex).
• The arms of the parabola extend downwards infinitely.

By finding where the vertex lies, we can determine the highest point on the curve—the location of maximum ozone density in this example. This geometric interpretation is fundamental in identifying important points like maximum and minimum in real-world applications.

Ozone Density

Ozone density refers to the concentration of ozone molecules in a given volume of the Earth's atmosphere, and it can vary by altitude, location, and time. At higher altitudes, especially between the stratosphere's 20 to 35 kilometers range, ozone plays a significant role in absorbing harmful ultraviolet radiation.

In Edmonton, Canada, ozone density was measured experimentally across various altitudes during different seasons, leading to a model like the one we analyzed: \(D(h) = -0.078h^2 + 3.811h - 32.433\) for spring. This quadratic model allows us to express the changes in ozone density in relation to altitude.

Understanding how to find the altitude with the maximum ozone density is crucial because at these peak levels, the protective role of ozone is maximized, shielding life on Earth from potential UV radiation damage.