Problem 37 - Applied Calculus

Question

An apple tree produces, on average, \(400 \mathrm{kg}\) of fruit each season. However, if more than 200 trees are planted per \(\mathrm{km}^{2},\) crowding reduces the yield by \(1 \mathrm{kg}\) for each tree over 200. (a) Express the total yield, \(y,\) from one square kilometer as a function of the number of trees on it. Graph this function. (b) How many trees should a farmer plant on each square kilometer to maximize yield?

Step-by-Step Solution

Answer

Plant 300 trees per square kilometer for maximum yield.

Step 1: Understand the Problem

We need to express the yield as a function of the number of trees per square kilometer, considering the production per tree and the yield reduction if the number of trees exceeds 200.

Step 2: Define Variables

Let \( n \) be the number of trees planted per square kilometer. The base yield per tree is 400 kg, but if \( n > 200 \), the yield per tree decreases by \( n - 200 \) kg.

Step 3: Express Yield Function

The yield per tree when more than 200 trees are planted is \( 400 - (n - 200) = 600 - n \). The total yield for \( n \) trees is thus \( y(n) = n(600 - n) \), simplifying to \( y(n) = 600n - n^2 \).

Step 4: Graph the Function

The function \( y(n) = 600n - n^2 \) is a downward-opening parabola (since the coefficient of \( n^2 \) is negative). The vertex of this parabola will give the maximum yield.

Step 5: Find Maximum Yield Number

To find the number of trees that maximizes yield, find the vertex of the parabola \( y = 600n - n^2 \). The vertex of a quadratic function \( ax^2 + bx + c \) is at \( x = -\frac{b}{2a} \). Here, \( a = -1, b = 600 \), so \( n = -\frac{600}{2(-1)} = 300 \).

Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions that follow the form \(ax^2 + bx + c\). These functions describe curved shapes when graphed, especially the parabola. They are commonly used in physics, finance, and engineering to model natural phenomena.
Quadratic functions have specific characteristics:

• They consist of three key parts: a quadratic (\(ax^2\)), a linear (\(bx\)), and a constant (\(c\)).
• The graph of a quadratic function is a parabola. Parabolas can open upwards or downwards.
• The highest or lowest point of this parabola is the "vertex".

In the context of the apple tree problem, the quadratic function is given by \(\ y(n) = 600n - n^2\). This is derived by determining how yield depends on the number of trees, incorporating the effects of crowding.
This function captures the relationship between tree density (\(n\)) and the total yield (\(y\)). The negative coefficient of \(n^2\) indicates a downward-opening parabola, meaning there’s a point where adding more trees will start to decrease the yield.

Maximization Problem

A maximization problem asks us to find the conditions that result in the largest possible value for a certain function. This is especially important in agriculture, economics, and manufacturing, where resources like space and labor are limited.
To solve a maximization problem:

• Define the function you need to maximize, which represents the thing you want more of (e.g., yield, profit).
• Identify constraints or conditions affecting this function.
• Use calculus or algebra (like vertex formula for parabolas) to find the input value that provides the maximum output.

In the apple tree scenario, the function \(y(n) = 600n - n^2\) models total yield based on tree numbers, and the task is to discover the optimal \(n\). Calculating the vertex, the peak of this function, reveals how many trees should be planted to achieve the maximum yield.

Vertex of a Parabola

The vertex of a parabola is an important feature. For a quadratic function \(ax^2 + bx + c\), the vertex can be found using the formula \(x = -\frac{b}{2a}\). It provides the maximum or minimum value of the function, depending on whether the parabola opens downwards or upwards.
Key points about the vertex:

• For downward-opening parabolas, like \(y(n) = 600n - n^2\), the vertex represents the maximum point.
• For upward-opening parabolas, the vertex indicates the minimum point.

Within the apple tree scenario, using \(n = -\frac{b}{2a}\) gives us \(n = 300\). This calculation tells us that planting 300 trees per square kilometer maximizes the yield. By understanding where the vertex lies, we can make optimal decisions for resource allocation or other similar tasks.