Problem 65 - 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 in order 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 given is \[ A(n) = n(900 - 9n)\] This function represents the total apples produced per acre based on the number of trees, \(n\). It is a quadratic function because it can be expanded to the form \[ A(n) = 900n - 9n^2\] which is a standard quadratic form with \(-9n^2\) as the leading term, causing the parabola to open downwards.

Step 2: Identify the Vertex of the Parabola

To find the number of trees that maximizes the yield, we need to find the vertex of the parabola.The formula to find the vertex \(n\) of a quadratic function \(an^2 + bn + c\) is given by:\[ n = -\frac{b}{2a}\]In our function, \(a = -9\), \(b = 900\). Plug in these values:\[ n = -\frac{900}{2(-9)} = \frac{900}{18} = 50\]So, 50 trees should be planted per acre for maximum yield.

Step 3: Verify the Result

We found that \(n = 50\) trees maximize the yield. To confirm, we can substitute back into the equation for verification or simply note that the calculated vertex corresponds to expected maximums in similar quadratic problems. Since the parabola opens downward and the formula used is correct, \(n = 50\) ensures the maximum yield without requiring recalculation here.

Key Concepts

Vertex of a Parabola

In a quadratic function, the vertex of a parabola is a crucial point that can often determine maximizing or minimizing outcomes. The vertex is the peak or lowest point of the parabola, depending on the orientation. The standard form of a quadratic equation is given by \( an^2 + bn + c \). Here, the vertex can be found using the formula \( n = -\frac{b}{2a} \), where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation.

In our scenario, the function representing apple yield per acre is \( A(n) = 900n - 9n^2 \). With the coefficients \( a = -9 \) and \( b = 900 \), the vertex lies where \( n = -\frac{900}{2(-9)} = 50 \). This calculation identifies the number of trees per acre to achieve optimal apple production. The vertex not only gives the number of trees but also represents the number of apples produced at this optimal planting density.

Optimization Problem

Optimization problems involve finding the best solution from a set of feasible options. Here, our task is to determine how many trees should be planted per acre to maximize apple yield. Such issues often rely on quadratic functions, where maximizing or minimizing the function reflects the highest or lowest value within a defined range.

The function \( A(n) = 900n - 9n^2 \) is typical of an optimization scenario. By completing or using the vertex formula of the quadratic function, we found that 50 trees should be planted to achieve the optimal apple yield per acre. Optimization offers benefits such as:

• Conserving resources by using land efficiently.
• Maximizing outputs for economic profitability.
• Aiding in planning for future expansions or adjustments.

Understanding optimization in quadratic functions is invaluable in practical applications like agriculture, economics, and engineering.

Maximum Yield

The maximum yield in this context represents the most apples that can be produced per acre, factoring in tree density. Achieving maximum yield is a balance between too many trees, which might stifle growth due to competition, and too few trees, failing to optimize the available land.

Our equation \( A(n) = 900n - 9n^2 \) reveals that planting 50 trees per acre optimizes the yield of apples. This result is derived by identifying the vertex of the downward-opening parabola. The chosen planting density leverages the peak point of the parabola, ensuring that each tree has enough space and resources to produce the maximum number of apples.

Consequently, accessing maximum yield involves strategic decisions around planting density, resource allocation, and environmental factors. These considerations significantly impact productivity and efficiency in agricultural practices. By understanding the mathematical foundation through the vertex of a parabola, farmers and orchard managers can effectively determine the best planting strategies for maximum yield.