Ozone occurs at all levels of the earth's atmosphere. The...

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

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

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Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial function typically expressed as ax² + bx + c. Its graph is a parabola, which can open upward or downward depending on the sign of the coefficient a. Quadratic functions are widely used in various fields to model phenomena with a peak or valley.

Vertex of a Parabola

The vertex of a parabola is the point where the function reaches its maximum or minimum value, depending on the orientation of the curve. In the quadratic equation ax² + bx + c, if a is negative, the vertex indicates the maximum point. The vertex can be found using the formula -b/(2a) for the x-coordinate, which is crucial in optimization problems.

Optimization in Quadratic Models

Optimization involves finding the maximum or minimum value of a function, which is often essential in practical applications such as determining peak values. By modeling a situation with a quadratic function and using the vertex formula, one can efficiently determine the point of greatest (or least) value, facilitating decision-making and analysis.

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