Calculus help please..........(orchard problem)?

Question

In an apple orchard there are 30 trees per acre, and the average yield per tree is 400 apples. For each additional tree planted per acre, the average yield per tree is reduced by 10 apples. How many trees per acre will maximize the crop?

What's the equation for yield per acre?

What's the derivative?

What's the optimum number of trees?

Answer 1

let t = number of trees in an acre

let y = yield per tree
y = 400 less 10 per tree over 30
y = 400 - ((t - 30) * 10)
y = 400 - 10t + 300
y = 700 - 10t

let a = yield per acre
a = yield per tree * trees per acre
a = y * t
a = (700 - 10t) * t
a = 700t - 10t^2

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The derivate of a, da/dt =

da/dt = d(700t - 10t^2)/dt
da/dt = 700 - 20t

derivative = 700 - 20t

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Optimum occurs when derivative = 0:

da/dt = 0
700 - 20t = 0
20t = 700
t = 35

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Now, let's check:

a = 700t - 10t^2

when t=30, a = 700*30 - 10*30*30 = 12,000
when t=34, a = 700*34 - 10*34*34 = 12,240
when t=35, a = 700*35 - 10*35*35 = 12,250
when t=36, a = 700*36 - 10*36*36 = 12,240
when t=40, a = 700*40 - 10*40*40 = 12,000

t=35 looks like a maximum to me.