The population P of a culture of P seudomonas aeruginosa...

The population $P$ of a culture of $P$ seudomonas aeruginosa bacteria is given by $P=-1718 t^{2}+$ $82,000 t+10,000,$ where $t$ is the time in hours since the culture was started. Determine the time(s) at which the population was 600,000 . Round to the nearest hour. (See Example 4)

Key Concepts

Quadratic Equations

A quadratic equation is an equation of the second degree, typically written in the form ax^2 + bx + c = 0, where a, b, and c are constants and a ? 0. Quadratic equations arise naturally in many contexts, including physics, economics, and biology, when modeling situations that involve acceleration or growth that changes over time.

Quadratic Formula

The quadratic formula, x = (-b ± ?(b² - 4ac)) / (2a), provides a method to find the roots of any quadratic equation. It is especially useful when the quadratic equation cannot be easily factored, ensuring that all real (and complex) solutions are identified. This formula also involves the discriminant (b² - 4ac), which determines the nature of the roots.

Rounding and Estimation

Rounding is the process of approximating a number to a desired degree of accuracy, which is essential when the exact value is impractical or unnecessary. In many real-world applications, such as time measurements, results are rounded to the nearest whole unit to align with how measurements are taken or reported. Estimation assists in validating the reasonableness of calculated values.

Interpreting Solutions in Context

When applying quadratic equations to solve real-world problems, it is important to interpret the solutions appropriately. This involves considering the physical context—such as the time at which an event occurs—where not all mathematical solutions may be applicable. Ensuring that the identified solutions make sense and meet any constraints (like non-negative time values) is critical for accurate interpretation.

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