The number of apples produced by each tree in an apple...

The number of apples produced by each tree in an apple orchard depends on how densely the trees are planted. If $n$ trees are planted on an acre of land, then each tree produces $900-9 n$ apples. So the number of apples produced per acre is $$ A(n)=n(900-9 n) $$ How many trees should be planted per acre to obtain the maximum yield of apples? (FIGURE CAN'T COPY)

The number of apples produced by each tree in an apple orchard depends on how densely the trees are planted. If $n$ trees are planted on an acre of land, then each tree produces $900-9 n$ apples. So the number of apples produced per acre is
$$
A(n)=n(900-9 n)
$$
How many trees should be planted per acre to obtain the maximum yield of apples?
(FIGURE CAN'T COPY)

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Key Concepts

Vertex of a Parabola

The vertex of a parabola is the point where the quadratic function attains its extreme value—either maximum or minimum. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex can be calculated using the formula -b/(2a). This concept is important because the vertex provides the optimal solution in many real-world scenarios involving quadratic models.

Optimization in Quadratic Models

Optimization refers to the process of finding the maximum or minimum value of a function, which in quadratic models is achieved by locating the vertex of the parabola. This technique is widely used in problems that involve maximizing or minimizing a particular quantity, such as yield, profit, or efficiency, by identifying the best potential conditions from the quadratic relationship.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. Its graph is a parabola, and the sign of the coefficient a determines whether the parabola opens upward (a > 0) or downward (a < 0), which in turn indicates whether the function has a minimum or maximum value, respectively.

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