Problem 70 - Precalculus Mathematics for Calculus

Question

At a certain vineyard it is found that each grape vine produces about 10 pounds of grapes in a season when about 700 vines are planted per acre. For each additional vine that is planted, the production of each vine decreases by about 1 percent. So the number of pounds of grapes produced per acre is modeled by $$A(n)=(700+n)(10-0.01 n)$$ where \(n\) is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.

Step-by-Step Solution

Answer

Plant 850 vines to maximize grape production per acre.

Step 1: Understanding the Function

The function given is \(A(n)=(700+n)(10-0.01n)\), where \(n\) is the number of additional vines planted. The goal is to find the value of \(n\) that maximizes the function, which represents the total grape production per acre.

Step 2: Expanding the Expression

To simplify the function, first expand the expression: \(A(n) = (700 + n)(10 - 0.01n) = 7000 + 10n - 7n - 0.01n^2 = 7000 + 3n - 0.01n^2\).

Step 3: Reformulating into Standard Quadratic Form

The expanded expression of the function is a quadratic equation: \(A(n) = -0.01n^2 + 3n + 7000\). This allows us to use calculus or the vertex formula to find the maximum point.

Step 4: Finding the Vertex

The formula for the vertex of a parabola given by \(ax^2 + bx + c\) is \(x = -\frac{b}{2a}\). Here, \(a = -0.01\), \(b = 3\). So, \(n = -\frac{3}{2(-0.01)} = 150\).

Step 5: Calculating the Optimal Number of Vines

The result from the vertex formula indicates that \(n = 150\). Since 700 was the initial number of vines, the total number of vines that should be planted is \(700 + 150 = 850\).

Key Concepts

Quadratic Function

In this vineyard problem, we are dealing with a quadratic function, which is a type of polynomial equation. The general form of a quadratic equation is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our specific scenario, the function \(A(n) = -0.01n^2 + 3n + 7000\) models the total grape production per acre based on the number of additional vines planted, \(n\).

Quadratic functions are characterized by their parabolic graphs. They exhibit symmetry and have a highest or lowest point known as a vertex. Depending on the leading coefficient, \(a\), the parabola opens upwards if \(a > 0\) or downwards if \(a < 0\). In our function, \(a = -0.01\), indicating that the parabola opens downward, and the vertex represents the maximum point of grape production.

Understanding how to read and manipulate quadratic functions is crucial in optimization problems like this, where you want to find the best conditions for maximum output.

Vertex Formula

The vertex formula is a powerful tool for finding the maximum or minimum point of a quadratic function. It is given by \(x = -\frac{b}{2a}\), where \(a\) and \(b\) are coefficients from the quadratic equation \(ax^2 + bx + c\). Using this formula, you can determine the value of \(x\) at which the function reaches its peak.

In our problem with the grape vine yield, the function derived was \(A(n) = -0.01n^2 + 3n + 7000\). Here, \(a = -0.01\) and \(b = 3\). Plugging these values into the vertex formula, we found \(n = -\frac{3}{2(-0.01)} = 150\). This calculation means that planting 150 additional vines will yield the maximum grape production.

The vertex formula simplifies the process of finding the optimal solution in quadratic problems, bypassing more complex calculus methods like finding derivatives. It's particularly helpful when you need a quick and efficient way to solve for maxima or minima.

Grape Vine Yield

Maximizing grape vine yield is essential for the profitability and efficiency of a vineyard. In this exercise, the yield is dependent on the number of vines planted per acre and how each additional vine affects production. Initially, each vine produces about 10 pounds of grapes when there are 700 vines per acre. However, for each additional vine planted, each vine's productivity decreases by approximately 1%.

This real-world scenario incorporates biological and environmental constraints into the mathematical model. The function \(A(n) = (700+n)(10-0.01n)\) is used to represent this situation. Expanding and simplifying this function helps identify the optimal planting strategy.

By determining the number of additional vines that maximizes the yield using our quadratic formula, we find that planting 850 vines in total (700 initially plus 150 additional) yields the highest total grape production. Understanding the balance between the number of vines and their yield is vital for vineyard operators looking to maximize their output effectively.