Problem 69 - Precalculus Mathematics for Calculus

Question

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 to obtain the maximum yield of apples.

Step 1: Define the Function

The number of apples produced per acre is given by the function \( A(n) = n(900 - 9n) \). This represents the total number of apples produced when \( n \) trees are planted per acre.

Step 2: Expand the Function

Expand the expression \( A(n) = n(900 - 9n) \) to get a quadratic equation: \[ A(n) = 900n - 9n^2 \] This is a quadratic function in the standard form \( A(n) = -9n^2 + 900n \).

Step 3: Identify the Coefficients

In the quadratic equation \( A(n) = -9n^2 + 900n \), identify the coefficients: - The coefficient of \( n^2 \) is \( a = -9 \) - The coefficient of \( n \) is \( b = 900 \)

Step 4: Find the Vertex of the Parabola

A quadratic function \( ax^2 + bx + c \) reaches its maximum or minimum at the vertex. The \( n \)-coordinate of the vertex of the parabola is given by the formula: \[ n = \frac{-b}{2a} \] Substitute for \( a = -9 \) and \( b = 900 \).

Step 5: Calculate the Maximum Tree Planting

Substitute the values into the vertex formula: \[ n = \frac{-900}{2(-9)} = \frac{-900}{-18} = 50 \] Thus, the maximum number of trees that should be planted per acre is \( 50 \).

Key Concepts

Maximizing Yield in Quadratic Functions

In the context of an apple orchard, maximizing yield means finding the ideal number of trees to plant per acre to produce the most apples. Given a situation where the yield per tree changes based on the number of trees, modeling this situation with a quadratic function is common.
The function that represents the yield, such as \( A(n) = n(900 - 9n) \), will show the total apple production as a result of planting \( n \) trees. Quadratic functions like this often form a parabolic curve when graphed. This curve will have a single maximum point, which indicates the optimal number of trees that maximize apple production.
To find this peak, or maximum yield, one must determine the vertex of the parabola. The vertex serves as the point where the curve achieves its highest value, indicating the planted number of trees that provides the highest yield.

Vertex Formula in Quadratic Equations

The vertex formula is a powerful tool in solving quadratic equations for either maximum or minimum values. This is especially useful in practical applications such as optimizing production yields.
When dealing with a quadratic equation in the standard form \( ax^2 + bx + c \), the coordinates of the vertex \((n, A(n))\) in the context of our problem, can be pinpointed using the formula:

• For the \( n \)-coordinate: \( n = \frac{-b}{2a} \).
• The \( A(n) \)-coordinate represents the maximum or minimum value of the function.

To apply this, plug the coefficients \( a \) and \( b \) from the expanded equation \( A(n) = -9n^2 + 900n \) into the formula. This formula simplifies the process of finding how many trees should be planted per acre to maximize the apple yield.

Quadratic Equations in Real-Life Applications

Quadratic equations are versatile mathematical tools often used to model real-world problems. They come into play when a situation involves a relationship that follows a parabolic pattern, having an increase, a peak, and then a decrease.
In the apples per acre situation described, the goal is to discover how planting more or fewer trees affects the total number of apples produced. The function \( A(n) = -9n^2 + 900n \) is set in such a way that it reflects how each additional tree impacts the total yield. This function can predict a point where increasing the number of trees will no longer add to, but may start decreasing the total yield.
Quadratic equations help us to visualize and compute optimal solutions in diverse fields like agriculture, economics, and engineering. Solving these equations allows for informed decision-making to maximize desired outcomes such as yield, efficiency, or profit.