1. ## SOLVING LINEAR FUNCTION

A farmer has 70-acres on which he wants to plant a pear orchard. Three neighboring farms with similar soil conditions already have established orchards. Based on the yield from the neighboring farms, the yield per tree appears to be a linear function of the tree density (trees per acre). Letting t be the number of trees planted per acre, the funtion P(t)= -1.5t + 900 represents the yield in pears per tree.

What function of t would then give the yield per acre for the orchard? Identify the vertex.

What is the optimal number of trees to plant per acre?

2. Originally Posted by nelle87
A farmer has 70-acres on which he wants to plant a pear orchard. Three neighboring farms with similar soil conditions already have established orchards. Based on the yield from the neighboring farms, the yield per tree appears to be a linear function of the tree density (trees per acre). Letting t be the number of trees planted per acre, the funtion P(t)= -1.5t + 900 represents the yield in pears per tree.

What function of t would then give the yield per acre for the orchard? Identify the vertex.

What is the optimal number of trees to plant per acre?
P(t) = -1.5t +900 is the pears per tree.

In one acre, there are are t number of trees planted.
So,
Y(t) = (-1.5t +900)*t would be the yield per acre.
Y(t) = -1.5t^2 +900t pears per acre --------------answer.

That is a vertical parabola that opens downward, because of the negative t^2. Its vertex then is a maximum point.
To find its vertex, by using the formula "x = -b/ 2a",
t = -900/ 2(-1.5) = 300
Meaning, when there are 300 trees planted per acre, the yield per acre is maximum.

At t=300,
Y(300) = -1.5(300)^2 +900(300) = 135,000 pears

So, the vertex is (300trees, 135,000pears) --------answer.
And so, the optimal number of trees to plant per acre is 300. ----answer.

Check:
Yield per tree, = -1.5(300) +900 = 450 pears per tree
Hence, yield per acre = (450)(300) = 135,000 pears per acre.
So, OK.