Problem 100 - Algebra and Trigonometry

Question

The annual yield per lemon tree is fairly constant at 320 pounds when the number of trees per acre is 50 or fewer. For each additional tree over \(50,\) the annual yield per tree for all trees on the acre decreases by 4 pounds due to overcrowding. Find the number of trees that should be planted on an acre to produce the maximum yield. How many pounds is the maximum yield?

Step-by-Step Solution

Answer

To produce the maximum yield, plant 65 trees on an acre. The maximum yield is 17550 pounds.

Step 1: Define the Mathematical Model

First, we need to establish a mathematical model to represent the problem. Let \(x\) be the number of trees exceeding 50. The total yield \(Y\) can be represented with this function: \(Y = (320 - 4x)(50 + x) = -4x^2 + (320 - 200)x + 16000\). This is a quadratic function, with a negative leading coefficient, indicating that the graph is a downward parabola. The maximum yield is the vertex of the parabola.

Step 2: Find the vertex of the parabola with formula

The vertex of parabola given by the standard quadratic equation \(ax^2+bx+c\) has the x-coordinate given by \(-b/2a\). For our function \(Y = -4x^2 + (320 - 200)x + 16000\), the vertex x-coordinate (i.e., number of trees that exceed 50) can be found with \(-b/2a = -(320 - 200) / (2*-4) = 15\).

Step 3: Determine the actual number of trees

Since \(x\) represents the number of trees exceeding 50, we need to add 50 to find the total number of trees that should be planted for maximum yield. So, the ideal number of trees is \(50 + x = 50 + 15 = 65\).

Step 4: Calculate the maximum yield

Substitute \(x\) back into the yield function \(Y\), to calculate the maximum yield in pounds. \(Y(15) = -4*(15)^2 + (320 - 200) * 15 + 16000 = 17550\)pounds.