Problem 14 - Precalculus Essentials

Question

The annual yield per orange tree is fairly constant at 270 pounds per tree when the number of trees per acre is 30 or fewer. For each additional tree over \(30,\) the annual yield per tree for all trees on the acre decreases by 3 pounds due to overcrowding. a. Express the yield per tree, \(Y\), in pounds, as a function of the number of orange trees per acre, \(x\). b. Express the total yield for an acre, \(T\), in pounds, as a function of the number of orange trees per acre, \(x\).

Step-by-Step Solution

Answer

The yield per tree, \(Y\), in pounds, as a function of the number of orange trees per acre, \(x\), can be expressed as \(Y(x) = \)\[ \begin{cases} 270 & \text{if } x \leq 30, \270 - 3(x - 30) & \text{if } x > 30.\end{cases}\]and the total yield for an acre, \(T\), in pounds, as a function of the number of orange trees per acre, \(x\), can be expressed as \(T(x) = \)\[ \begin{cases} 270x & \text{if } x \leq 30, \(270 - 3(x - 30)) \cdot x & \text{if } x > 30.\end{cases}\]

Step 1: Express the yield per tree, \(Y\), in pounds, as a function of the number of orange trees per acre, \(x\).

The yield per tree depends on the number of trees per acre. If the number of trees is 30 or less, then each tree produces 270 pounds. However, if the number of trees increases above 30, each tree produces 3 pounds less for each additional tree. Therefore, the yield can be represented as follows: \(Y(x) = \) \[ \begin{cases} 270 & \text{if } x \leq 30, \270 - 3(x - 30) & \text{if } x > 30.\end{cases}\]

Step 2: Express the total yield for an acre, \(T\), in pounds, as a function of the number of orange trees per acre, \(x\).

The total yield for an acre will be the yield per tree times the number of trees. Using the \(Y(x)\) function from the previous step, the total yield can be calculated as: \(T(x) = x \cdot Y(x)\). Therefore, the function can be represented as follows: \(T(x) = \)\[ \begin{cases} 270x & \text{if } x \leq 30, \(270 - 3(x - 30)) \cdot x & \text{if } x > 30.\end{cases}\]

Key Concepts

Piecewise Functions

In mathematics, piecewise functions are crucial for situations where a function has different expressions based on the input values. This is exactly the case with the yield per orange tree. When the number of trees, denoted as \(x\), is 30 or fewer, the yield remains constant at 270 pounds. Beyond 30 trees, each additional tree leads to a decrease in yield because of the overcrowding effect. Here's how the function looks for the yield per tree, \(Y(x)\):

• If \(x \leq 30\), then \(Y(x) = 270\)
• If \(x > 30\), then \(Y(x) = 270 - 3(x - 30)\)

This piecewise function allows us to easily calculate the yield per tree based on how overloaded the acre is with trees. It breaks down a complex problem into manageable pieces and clearly defines the conditions under which each expression is valid.

Overcrowding Effect

The overcrowding effect is a common problem in agriculture, where adding too many trees or plants within a certain area diminishes the overall yield. This happens because resources like nutrients, water, and sunlight become limited as more plants compete for them. In the context of our problem, each additional tree beyond the initial 30 reduces the yield by 3 pounds per tree. This is because the trees start competing for the same resources, which leads to reduced growth and yield. Managing overcrowding is crucial as it directly affects the productivity of the farm. It's important to understand that the goal is not to plant as many trees as possible but to find an optimal number where the yield per tree doesn't drop significantly due to overcrowding. It showcases the delicate balance between resource management and agricultural yield.

Agricultural Yield Optimization

Optimizing agricultural yield involves finding the right balance between the number of trees and the yield per tree. Here, we're given a formula for total yield, \(T(x)\), which factors in the number of trees and their respective yield:

• If \(x \leq 30\), then \(T(x) = 270x\)
• If \(x > 30\), then \(T(x) = (270 - 3(x - 30)) \cdot x\)

The aim of yield optimization is to maximize the total yield while avoiding the negative impacts of overcrowding. This involves using both mathematical modeling and experimental insight to predict how changes in planting density will impact the yield. It's vital to consider external factors such as soil quality and climate conditions when determining the ideal density for planting.Effective optimization ensures that the land is used effectively, leading to higher productivity and profitability in farming operations. It is a practical application of mathematical principles to real-world scenarios, aiding farmers in decision-making.