Question

A peach orchard owner wants to maximize the amount of peaches produced by her orchard.

She has found that the per-tree yield is equal to 900 whenever she plants 30 or fewer trees per acre, and that when more than 30 trees are planted per acre, the per-tree yield decreases by 40 peaches per tree for every extra tree planted.
For example, if there were 25 trees planted per acre, each tree would produce 900 peaches. If there were 35 trees planted per acre, each tree would produce 900 - 40 * (35 - 30) peaches, which is roughly equal to 700 peaches.
a.Find the function that describes the per-tree yield, Y, in terms of x.
Y = if x is no more than 30 trees per acre
Y = if x is greater than 30 trees per acre
b.Find the total yield per acre, T, that results from planting x trees per acre.
T = if x is no more than 30 trees per acre
T = if x is greater than 30 trees per acre
c.Differentiate T with respect to x
dT / dx = if x is less than 30 trees per acre
dT / dx = if x is greater than 30 trees per acre

Answer 1

a) Y(x) = {900, x≤30; 900-40(x-30), x>30}

b) T(x) = {900x, x≤30; 2100x-40x², x>30}

c) dT/dx = {900, x≤30; 2100-80x, x>30}

Explanation:

a) The problem statement gives the function for x ≤ 30, and gives an example of evaluating the function for x = 35. So, replacing 35 in the example with x gives the function definition for x > 30.

`\displaystyle Y(x)=\left\{\begin{array}{lcl}900&\text{for}&x\le 30\\900-40(x-30)&\text{for}&x>30\end{array}\right.`

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b) The yield per acre is the product of the number of trees and the yield per tree:

T(x) = x·Y(x)

`\displaystyle T(x)=\left \{\begin{array}{lcl}900x&\text{for}&x\le 30\\2100x-40x^2& \text{for}&x>30\end{array}\right.`

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c) The derivative is ...

`\displaystyle(dT)/(dx)=\left \{\begin{array}{lcl}900&\text{for}&x\le 30\\2100-80x& \text{for}&x>30\end{array}\right.`

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The attached graph shows the yield per acre (purple, overlaid by red for x<30), the total yield (black), and the derivative of the total yield (red). You will note the discontinuity in the derivative at x=30, where adding one more tree per acre suddenly makes the rate of change of yield be negative.