Question

Orchard Farming An apple farmer currently has 156 trees yielding an average of 12 bushels of apples per tree. He is expanding his farm at a rate of 13 trees per year, while improved husbandry is improving his average annual yield by 1.5 bushels per tree. What is the current (instantaneous) rate of increase of his total annual production of apples? Answer in appropriate units of measure.

Short answer

The current rate of increase of total annual apple production is 262.5 bushels per year.

Step-by-step solution

Define the variables

Let's denote the number of trees as \(T(t)\) and the yield per tree as \(Y(t)\). Current values are \(T(t) = 156\) trees and \(Y(t) = 12\) bushels/tree. Rates of change are given as \(\frac{dT}{dt} = 13\) trees/year and \(\frac{dY}{dt} = 1.5\) bushels/tree/year. The total output, \(P(t)\), is \( T(t) \cdot Y(t)\).

Differentiate the output function

We're looking for \(\frac{dP}{dt}\), the rate of change of the total annual production. Product rule of differentiation gives: \(\frac{dP}{dt} = T(t) \cdot \frac{dY}{dt} + Y(t) \cdot \frac{dT}{dt}\).

Substitute the known values

Substitute the known function values and their derivatives into the equation from Step 2: \(\frac{dP}{dt} = 156 \cdot 1.5 + 12 \cdot 13\).