A peach orchard owner wants to maximize the amount of peache

A peach orchard owner wants to maximize the amount of peaches produced by her orchard. She cannot simply plant as many trees as she can, since planting more trees will decrease the amount of fruit that each tree produces (the yield of each tree). She has found that the per-tree yield can be described by the equation Y = 800 - 30 x. Here Y Is the yield per tree and x is the number of trees planted per acre. For example, if there were 10 trees planted per acre, each tree would produce 800 - 30 * 10 = 500 peaches. Find the number of trees per acre that should be planted in order to produce the maximum crop and the resulting total yield. Number of trees per acre: Total yield: Give your first answer correct to two decimal places, and the second answer correct to the nearest peach per acre.

Solution

Y = 800 -3x , Y is yield per tree and x is no. of trees planted

So, Total Yield = Y*x = x(800 -3x) = -3x^2 +800x

IN order to maxmize the total yield :

ax^2 +bx +c =0

Vertex : x= -b/2a (maximum value)

x = -(800/(2*-3) ) = 800/6 = 400/3 = 133.33 no. of trees

Total yield =  -3(133.33)^2 +800*133.33 = 53333 paeches per acre

The answer seem to be correct