Question

Two years ago your orange orchard contained 70 trees, and the yield per tree was 70 bags of oranges. Last year you removed 10 of the trees and noticed that the yield per tree increased to 75 bags. Assuming that the yield per tree depends linearly on the number of trees in the orchard, what should you do this year to maximize your total yield?.

Answer 1

The number of trees that maximizes the yield is 105, so he should plant another 45 trees.

How to maximize the yield?

We know that the yield depends linearly on the number of trees. Remember that a linear relation is written as:

y = a*x + b

Where a is the slope and b is the y-intercept.

And we know that if a line passes through two points (x₁, y₁) and (x₂, y₂), then the slope can be written as:

$a = \frac{y_2 - y_1}{x_2 - x_1}$

Here we have the two points:

(70 trees, 70 bags)

(60 trees, 75 bags)

Then the slope is:

$a = \frac{7 bags - 75 bags}{70 trees - 60 trees} = -0.5 bags/tree$

Then the linear equation is, ignoring the units, something like:

y = -0.5*x + b

To find the value of b we can use one of the known points, for example, the first one.

(70, 70)

70 = -0.5*70 + b

70 = -35 + b

70 + 35 = b = 105

Then the linear equation is:

y = -0.5*x + 105

This gives the yield per tree, and the total yield for the x trees will be x times the above quantity, so we define the total yield as:

T = -0.5x^2 + 105*x

Notice that this is a quadratic equation with a negative leading coefficient, meaning that the maximum is at the vertex.

And for a general quadratic function a*x^2 + b*x + c the vertex is at:

x = -b/2a

So in our case, the vertex is at:

x = -105/(2*(-0.5)) = 105

So to maximize the total yield, he should plant 45 more trees, to get a total of 105 trees.

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