1. ## Apple Orchard

Don't try to find an algebraic rule !!

the figure below is the graph of the annual yield , y(t), in bushels per year , from an orchard t years after planting. The trees take about 10 years to get established , but for the next 20 years they give a subtantial yield. After about 30 years , however, age and disease star to take their toll, and the annual yield falls off. ( see attachment for details)

* When should the orchard be cut down and replanted? assume that we want to maximize average revenue per year , and that fruit prices remain constant , so that this is achieved by maximizing average annual yield . use the graph of y(t) to estimate the time at which the average annual yield is a maximum. Explain your answer geometrically and symbolically.

well, I know that y(t) is the total annual yield . I used the second fundamental theorem of calculus to find a function for F(M), which is $F(M)= \int_{0}^{M}y(t)dt$ . The graph of F(M) is increasing concave up from 0 to 33 , and then it's increasing concave down .

I also found an expression for the average annual yield $\frac{1}{M}\int_{0}^{M} y(t)dt$

Based on the graph of F(M) , I believe the orchard should be cut down approximately after 50 years .However, I'm still confused about maximizing the average annnual yield using the graph. Can someone help me with this exercise? Thanks in advance!

2. To maximize the average annual yield, first get an expression for it so as to get intuition. So solve to find the point M where the derivative of the average annual yield is 0.
Doing so, I get that point occurs when
$\displaystyle y(M) =\frac{1}{M}\int_{0}^{M} y(t)dt$

at this point it'll be helpful to denote the right hand side of this equation, the average annual yield, as AAY.

Now looking at the graph, when M ~ 33 years, y(33)=33*500 > AAY, since the integrand in the expression for AAY would have to be a constant for them to have a chance to be equal. In partiuclar, y(M)>AAY for all values of M<33 years. This means we need to go out further in time, so as to add more area to AAY, while y(M) itself decreases since the function is concave down. 50 years seems reasonable.