Problem 69 - Algebra and Trigonometry

Question

Agriculture 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?

Step-by-Step Solution

Answer

Plant 50 trees per acre for maximum yield.

Step 1: Understand the Function

The function \( A(n) = n(900 - 9n) \) represents the total number of apples produced per acre, where \( n \) is the number of trees planted per acre. This is a quadratic function in the standard form \( A(n) = an^2 + bn + c \).

Step 2: Express in Standard Quadratic Form

Expand the function: \[ A(n) = n(900 - 9n) = 900n - 9n^2 \]. Thus, the quadratic function is \( A(n) = -9n^2 + 900n \), where \(a = -9\), \(b = 900\).

Step 3: Find the Vertex

The maximum or minimum of a quadratic function \(ax^2 + bx + c\) is at \(x = -\frac{b}{2a}\). Here, \(a = -9\) and \(b = 900\). Substitute these values to find the vertex: \[ n = -\frac{900}{2(-9)} = -\frac{900}{-18} = 50 \].

Step 4: Verify the Maximum Point

Since \(a = -9 < 0\), the parabola opens downwards. This means that the vertex represents a maximum point.

Key Concepts

Quadratic Functions

One of the most common mathematical patterns you'll encounter is the quadratic function. Quadratic functions are polynomial functions of degree two. They take the form: \[ f(x) = ax^2 + bx + c \] where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The graph of a quadratic function is a curve called a parabola. This smooth, U-shaped figure can open upwards or downwards depending on the sign of \( a \). If \( a \) is positive, the parabola opens upwards, creating a minimum point at its vertex. Conversely, if \( a \) is negative, the parabola opens downwards, indicating a maximum point at the vertex. Quadratic functions model a variety of real-life situations such as projectile motion, areas, and, as in this example, optimizing the yield in agriculture. By understanding how to manipulate and analyze quadratics, you can solve numerous practical problems efficiently.

Vertex of a Parabola

The vertex of a parabola plays a crucial role in understanding the quadratic function. It is the point where the parabola changes direction, and it can either be the lowest or the highest point on the graph, depending on whether the parabola opens upwards or downwards. The vertex formula helps us find this point: \[ x = -\frac{b}{2a} \] where \( a \) and \( b \) are the coefficients from the quadratic equation \( ax^2 + bx + c \). Once you find the \( x \)-value of the vertex, substitute it back into the function to get the \( y \)-value, creating the coordinate \( (x, f(x)) \). This vertex is especially important when you are tasked with maximizing or minimizing a quadratic function, as it provides you with an efficient way to find the extremum point without needing to plot the entire graph. In our example, this means determining the optimal number of trees to plant for maximum apple yield.

Parabola Properties

The shape and orientation of a parabola are determined by the properties derived from the quadratic function. These include the direction it opens, the width or "steepness," and its symmetry. Here are some key parabola properties to remember:

• Opening Direction: Determined by the sign of \( a \). Positive \( a \) opens upwards; negative \( a \) opens downwards.
• Axis of Symmetry: The vertical line that passes through the vertex, calculated using \( x = -\frac{b}{2a} \). It divides the parabola into two mirror-image halves.
• Maximum or Minimum Point: Located at the vertex. This is where the function's value reaches its peak or lowest point, crucial for solving optimization problems.
• Width: Determined by the absolute value of \( a \). A larger absolute \( a \) compresses the parabola, while a smaller absolute \( a \) makes it wider.

These properties allow you to predict the behavior of the parabola and use it to solve problems effectively, such as determining maximum outputs or optimal solutions in various fields.