Daily oil production by Pemex, Mexico's national oil...

Daily oil production by Pemex, Mexico's national oil company, for 2001-2009 can be approximated by $$ P(t)=-0.022 t^{2}+0.2 t+2.9 $$ million barrels $/$ day $(1 \leq t \leq 9)$ where $t$ is time in years since the start of $2000 .{ }^{3}$ According to the model, in what year was oil production by Pemex greatest? How many barrels per day were produced that year?

Daily oil production by Pemex, Mexico's national oil company, for 2001-2009 can be approximated by
$$
P(t)=-0.022 t^{2}+0.2 t+2.9
$$
million barrels $/$ day $(1 \leq t \leq 9)$ where $t$ is time in years since the start of $2000 .{ }^{3}$ According to the model, in what year was oil production by Pemex greatest? How many barrels per day were produced that year?

Key Concepts

Domain Restrictions

Domain restrictions define the interval over which a function is analyzed. In real-world applications, the model may only be valid or meaningful within a specific interval. Recognizing these boundaries is critical as it ensures that the optimization and resulting conclusions are practically applicable within the defined time frame.

Vertex of a Parabola

The vertex of a parabola is the point where the function attains its maximum or minimum value. For a quadratic function, the vertex can be found using the formula t = -b/(2a). When the parabola opens downward (a < 0), the vertex represents the maximum value, which is key in determining the highest point in optimization problems.

Quadratic Function

A quadratic function is a second-degree polynomial of the form ax² + bx + c. It commonly represents parabolic behavior, where the sign of the coefficient a determines whether the parabola opens upward or downward. This concept is fundamental in modeling and analyzing situations where the rate of change itself changes at a constant rate.

Optimization

Optimization involves finding the maximum or minimum value of a function within a given domain. In the context of a quadratic function, the vertex provides the optimal value. This concept is essential in applied mathematics for scenarios like determining the highest production rate or maximum efficiency.

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