will you help me with grade 11 math?

Question

A farmer has data indicating that if he plants 30 orange trees per acre, each tree will yield an average of 360 oranges. For every additional tree planted, 15 fewer oranges will be produced per tree. How many trees per acre should the farmer plant to maximize his total crop? State both.

Answer 1

number of oranges = number of trees times number of oranges per tree, so let x = number of additional trees, 15x number of oranges less per tree:

N = (30+x)(360 - 15x)

now that's a parabola that opens down, crossing the x-axis at
0 = (30 + x)(360 - 15x)
0 = 30 + x
x = -30 ....... which means if he decreases the number of trees per acre by 30, he'll plant 0 trees and yield 0 oranges.
OR
0 = 360 - 15x
15x = 360
x = 24 ....... which means if he crowds 30 + 24 = 54 trees on each acre, again his yield will be 0.

the key here is that the vertex of the parabola, his max yield, is HALFWAY between the x-intercepts, so

(-30 + 24)/2 = -3
He'll maximize yield if plants only 27 trees per acre, getting 10,935 oranges per acre.

Answer 2

Answer: 10800 oranges as worked out below.

With 30 orange trees total yield = 30 * 360 = 10800 oranges.
With 31 orange trees total yield = 31 * 345 = 10695 oranges.
With 32 orange trees total yield = 32 * 330 = 10560.

As it is not given that with each tree less planted, 15 more oranges will be produced, maximum yield is 10800 oranges. Here, calculus will give an incorrect figure as it will consider more oranges per tree for each less tree.

I differ from the answer given by Philo which can be worked out in fewer steps as it would mean 27 trees to be planted and involve an assumption that for three less trees 45 more oranges will be the average yield which will be 27 * (360 + 45) = 10935. I understand from the language of the problem that the average yield remains the same as 360 upto 30 orange trees planted.