5. You have inherited an orange grove. The more orange trees there are in the grove, the fewer oranges each tree produces. When there are 30 trees, each tree produces 84 kg of oranges. When there are 90 trees, each tree produces 74 kg of oranges. Assume that the amount of oranges each tree produces is a linear function of the number of trees.

Question

1. Let x be the number of trees grown in the grove. In terms of x, what is the total number of kg of oranges produced on each tree?
2. In terms of x, what is the total number of kg of oranges produced in the grove?
3. What is the maximum number of kg of oranges that can be produced in the grove?

Answer 1

x y
30 84
90 74
So, every extra 60 trees reduces the yield by 10kg, so

y = -(10/60)x + c
Since y(30)=84,
(-1/6)(30)+c = 84
c=89

y = -x/6 + 89

The total yield is trees * avgyield, so

x(-x/6+89) = -x^2/6 + 89x

This is just a parabola, where the vertex is at x = 3*89

You can figure the total yield for that value of x.